:: SPRECT_1 semantic presentation
begin
registration
cluster
non
empty
V19
()
V22
(
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
Function-like
constant
V32
()
FinSequence-like
FinSubsequence-like
for ( ( ) ( )
set
) ;
end;
theorem
:: SPRECT_1:1
for
f
,
g
being ( (
V19
()
Function-like
FinSequence-like
) (
V19
()
V22
(
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
Function-like
V32
()
FinSequence-like
FinSubsequence-like
)
FinSequence
) st
f
: ( (
V19
()
Function-like
FinSequence-like
) (
V19
()
V22
(
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
Function-like
V32
()
FinSequence-like
FinSubsequence-like
)
FinSequence
)
^
g
: ( (
V19
()
Function-like
FinSequence-like
) (
V19
()
V22
(
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
Function-like
V32
()
FinSequence-like
FinSubsequence-like
)
FinSequence
) : ( (
V19
()
Function-like
FinSequence-like
) (
V19
()
V22
(
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
Function-like
V32
()
FinSequence-like
FinSubsequence-like
)
set
) is
constant
holds
(
f
: ( (
V19
()
Function-like
FinSequence-like
) (
V19
()
V22
(
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
Function-like
V32
()
FinSequence-like
FinSubsequence-like
)
FinSequence
) is
constant
&
g
: ( (
V19
()
Function-like
FinSequence-like
) (
V19
()
V22
(
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
Function-like
V32
()
FinSequence-like
FinSubsequence-like
)
FinSequence
) is
constant
) ;
theorem
:: SPRECT_1:2
for
x
,
y
being ( ( ) ( )
set
) st
<*
x
: ( ( ) ( )
set
) ,
y
: ( ( ) ( )
set
)
*>
: ( ( ) ( non
empty
non
trivial
V19
()
V22
(
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
Function-like
V32
() 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
FinSubsequence-like
)
set
) is
constant
holds
x
: ( ( ) ( )
set
)
=
y
: ( ( ) ( )
set
) ;
theorem
:: SPRECT_1:3
for
x
,
y
,
z
being ( ( ) ( )
set
) st
<*
x
: ( ( ) ( )
set
) ,
y
: ( ( ) ( )
set
) ,
z
: ( ( ) ( )
set
)
*>
: ( ( ) ( non
empty
non
trivial
V19
()
V22
(
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
Function-like
V32
() 3 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
FinSubsequence-like
)
set
) is
constant
holds
(
x
: ( ( ) ( )
set
)
=
y
: ( ( ) ( )
set
) &
y
: ( ( ) ( )
set
)
=
z
: ( ( ) ( )
set
) &
z
: ( ( ) ( )
set
)
=
x
: ( ( ) ( )
set
) ) ;
begin
theorem
:: SPRECT_1:4
for
GX
being ( ( non
empty
TopSpace-like
) ( non
empty
TopSpace-like
)
TopSpace
)
for
A
being ( ( ) ( )
Subset
of )
for
B
being ( ( non
empty
) ( non
empty
)
Subset
of ) st
A
: ( ( ) ( )
Subset
of )
is_a_component_of
B
: ( ( non
empty
) ( non
empty
)
Subset
of ) holds
A
: ( ( ) ( )
Subset
of )
<>
{}
: ( ( ) (
empty
trivial
Function-like
functional
FinSequence-membered
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
V245
() )
set
) ;
theorem
:: SPRECT_1:5
for
GX
being ( ( ) ( )
TopStruct
)
for
A
,
B
being ( ( ) ( )
Subset
of ) st
A
: ( ( ) ( )
Subset
of )
is_a_component_of
B
: ( ( ) ( )
Subset
of ) holds
A
: ( ( ) ( )
Subset
of )
c=
B
: ( ( ) ( )
Subset
of ) ;
theorem
:: SPRECT_1:6
for
T
being ( ( non
empty
TopSpace-like
) ( non
empty
TopSpace-like
)
TopSpace
)
for
A
being ( ( non
empty
) ( non
empty
)
Subset
of )
for
B1
,
B2
,
S
being ( ( ) ( )
Subset
of ) st
B1
: ( ( ) ( )
Subset
of )
is_a_component_of
A
: ( ( non
empty
) ( non
empty
)
Subset
of ) &
B2
: ( ( ) ( )
Subset
of )
is_a_component_of
A
: ( ( non
empty
) ( non
empty
)
Subset
of ) &
S
: ( ( ) ( )
Subset
of )
is_a_component_of
A
: ( ( non
empty
) ( non
empty
)
Subset
of ) &
B1
: ( ( ) ( )
Subset
of )
\/
B2
: ( ( ) ( )
Subset
of ) : ( ( ) ( )
Element
of
bool
the
carrier
of
b
1
: ( ( non
empty
TopSpace-like
) ( non
empty
TopSpace-like
)
TopSpace
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) )
=
A
: ( ( non
empty
) ( non
empty
)
Subset
of ) & not
S
: ( ( ) ( )
Subset
of )
=
B1
: ( ( ) ( )
Subset
of ) holds
S
: ( ( ) ( )
Subset
of )
=
B2
: ( ( ) ( )
Subset
of ) ;
theorem
:: SPRECT_1:7
for
T
being ( ( non
empty
TopSpace-like
) ( non
empty
TopSpace-like
)
TopSpace
)
for
A
being ( ( non
empty
) ( non
empty
)
Subset
of )
for
B1
,
B2
,
C1
,
C2
being ( ( ) ( )
Subset
of ) st
B1
: ( ( ) ( )
Subset
of )
is_a_component_of
A
: ( ( non
empty
) ( non
empty
)
Subset
of ) &
B2
: ( ( ) ( )
Subset
of )
is_a_component_of
A
: ( ( non
empty
) ( non
empty
)
Subset
of ) &
C1
: ( ( ) ( )
Subset
of )
is_a_component_of
A
: ( ( non
empty
) ( non
empty
)
Subset
of ) &
C2
: ( ( ) ( )
Subset
of )
is_a_component_of
A
: ( ( non
empty
) ( non
empty
)
Subset
of ) &
B1
: ( ( ) ( )
Subset
of )
\/
B2
: ( ( ) ( )
Subset
of ) : ( ( ) ( )
Element
of
bool
the
carrier
of
b
1
: ( ( non
empty
TopSpace-like
) ( non
empty
TopSpace-like
)
TopSpace
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) )
=
A
: ( ( non
empty
) ( non
empty
)
Subset
of ) &
C1
: ( ( ) ( )
Subset
of )
\/
C2
: ( ( ) ( )
Subset
of ) : ( ( ) ( )
Element
of
bool
the
carrier
of
b
1
: ( ( non
empty
TopSpace-like
) ( non
empty
TopSpace-like
)
TopSpace
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) )
=
A
: ( ( non
empty
) ( non
empty
)
Subset
of ) holds
{
B1
: ( ( ) ( )
Subset
of ) ,
B2
: ( ( ) ( )
Subset
of )
}
: ( ( ) ( non
empty
)
set
)
=
{
C1
: ( ( ) ( )
Subset
of ) ,
C2
: ( ( ) ( )
Subset
of )
}
: ( ( ) ( non
empty
)
set
) ;
begin
theorem
:: SPRECT_1:8
for
p
,
q
,
r
being ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Point
of ( ( ) ( non
empty
)
set
) ) holds
L~
<*
p
: ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Point
of ( ( ) ( non
empty
)
set
) ) ,
q
: ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Point
of ( ( ) ( non
empty
)
set
) ) ,
r
: ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Point
of ( ( ) ( non
empty
)
set
) )
*>
: ( ( ) ( non
empty
non
trivial
V19
()
V22
(
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
V23
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V32
() 3 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
FinSubsequence-like
)
FinSequence
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( )
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) )
=
(
LSeg
(
p
: ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Point
of ( ( ) ( non
empty
)
set
) ) ,
q
: ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Point
of ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) ( non
empty
compact
V258
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) ) : ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) ) )
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) )
\/
(
LSeg
(
q
: ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Point
of ( ( ) ( non
empty
)
set
) ) ,
r
: ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Point
of ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) ( non
empty
compact
V258
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) ) : ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) ) )
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( non
empty
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) ) ;
registration
let
n
be ( ( ) (
ordinal
natural
V11
()
real
ext-real
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) ) ;
let
f
be ( ( non
trivial
) ( non
empty
non
trivial
V19
()
V22
(
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
V23
( the
carrier
of
(
TOP-REAL
n
: ( ( ) (
ordinal
natural
V11
()
real
ext-real
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V32
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of ( ( ) ( non
empty
)
set
) ) ;
cluster
L~
f
: ( ( non
trivial
) ( non
empty
non
trivial
V19
()
V22
(
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
V23
( the
carrier
of
(
TOP-REAL
n
: ( ( ) (
ordinal
natural
V11
()
real
ext-real
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V32
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of the
carrier
of
(
TOP-REAL
n
: ( ( ) (
ordinal
natural
V11
()
real
ext-real
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( )
Element
of
bool
the
carrier
of
(
TOP-REAL
n
: ( ( ) (
ordinal
natural
V11
()
real
ext-real
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) )
->
non
empty
;
end;
registration
let
f
be ( ( ) (
V19
()
V22
(
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
V23
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V32
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of ( ( ) ( non
empty
)
set
) ) ;
cluster
L~
f
: ( ( ) (
V19
()
V22
(
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
V23
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V32
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( )
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) )
->
compact
;
end;
theorem
:: SPRECT_1:9
for
A
,
B
being ( ( ) ( )
Subset
of ) st
A
: ( ( ) ( )
Subset
of )
c=
B
: ( ( ) ( )
Subset
of ) &
B
: ( ( ) ( )
Subset
of ) is
horizontal
holds
A
: ( ( ) ( )
Subset
of ) is
horizontal
;
theorem
:: SPRECT_1:10
for
A
,
B
being ( ( ) ( )
Subset
of ) st
A
: ( ( ) ( )
Subset
of )
c=
B
: ( ( ) ( )
Subset
of ) &
B
: ( ( ) ( )
Subset
of ) is
vertical
holds
A
: ( ( ) ( )
Subset
of ) is
vertical
;
registration
cluster
R^2-unit_square
: ( ( ) ( non
empty
non
trivial
compact
special_polygonal
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) )
->
special_polygonal
non
horizontal
non
vertical
;
end;
registration
cluster
non
empty
compact
non
horizontal
non
vertical
for ( ( ) ( )
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) ) ;
end;
begin
theorem
:: SPRECT_1:11
for
C
being ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of ) holds
(
N-min
C
: ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of ) : ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
in
C
: ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of ) &
N-max
C
: ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of ) : ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
in
C
: ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of ) ) ;
theorem
:: SPRECT_1:12
for
C
being ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of ) holds
(
S-min
C
: ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of ) : ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
in
C
: ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of ) &
S-max
C
: ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of ) : ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
in
C
: ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of ) ) ;
theorem
:: SPRECT_1:13
for
C
being ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of ) holds
(
W-min
C
: ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of ) : ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
in
C
: ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of ) &
W-max
C
: ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of ) : ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
in
C
: ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of ) ) ;
theorem
:: SPRECT_1:14
for
C
being ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of ) holds
(
E-min
C
: ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of ) : ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
in
C
: ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of ) &
E-max
C
: ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of ) : ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
in
C
: ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of ) ) ;
theorem
:: SPRECT_1:15
for
C
being ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of ) holds
(
C
: ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of ) is
vertical
iff
W-bound
C
: ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of ) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) )
=
E-bound
C
: ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of ) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) ) ) ;
theorem
:: SPRECT_1:16
for
C
being ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of ) holds
(
C
: ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of ) is
horizontal
iff
S-bound
C
: ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of ) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) )
=
N-bound
C
: ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of ) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) ) ) ;
theorem
:: SPRECT_1:17
for
C
being ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of ) st
NW-corner
C
: ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of ) : ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
=
NE-corner
C
: ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of ) : ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) holds
C
: ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of ) is
vertical
;
theorem
:: SPRECT_1:18
for
C
being ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of ) st
SW-corner
C
: ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of ) : ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
=
SE-corner
C
: ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of ) : ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) holds
C
: ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of ) is
vertical
;
theorem
:: SPRECT_1:19
for
C
being ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of ) st
NW-corner
C
: ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of ) : ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
=
SW-corner
C
: ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of ) : ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) holds
C
: ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of ) is
horizontal
;
theorem
:: SPRECT_1:20
for
C
being ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of ) st
NE-corner
C
: ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of ) : ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
=
SE-corner
C
: ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of ) : ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) holds
C
: ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of ) is
horizontal
;
theorem
:: SPRECT_1:21
for
C
being ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of ) holds
W-bound
C
: ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of ) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) )
<=
E-bound
C
: ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of ) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) ) ;
theorem
:: SPRECT_1:22
for
C
being ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of ) holds
S-bound
C
: ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of ) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) )
<=
N-bound
C
: ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of ) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) ) ;
theorem
:: SPRECT_1:23
for
C
being ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of ) holds
LSeg
(
(
SE-corner
C
: ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of )
)
: ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) ,
(
NE-corner
C
: ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of )
)
: ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) ( non
empty
compact
V258
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) ) : ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) ) )
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) )
=
{
p
: ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Point
of ( ( ) ( non
empty
)
set
) ) where
p
is ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Point
of ( ( ) ( non
empty
)
set
) ) : (
p
: ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Point
of ( ( ) ( non
empty
)
set
) )
`1
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) )
=
E-bound
C
: ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of ) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) ) &
p
: ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Point
of ( ( ) ( non
empty
)
set
) )
`2
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) )
<=
N-bound
C
: ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of ) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) ) &
p
: ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Point
of ( ( ) ( non
empty
)
set
) )
`2
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) )
>=
S-bound
C
: ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of ) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) ) )
}
;
theorem
:: SPRECT_1:24
for
C
being ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of ) holds
LSeg
(
(
SW-corner
C
: ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of )
)
: ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) ,
(
SE-corner
C
: ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of )
)
: ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) ( non
empty
compact
V258
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) ) : ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) ) )
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) )
=
{
p
: ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Point
of ( ( ) ( non
empty
)
set
) ) where
p
is ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Point
of ( ( ) ( non
empty
)
set
) ) : (
p
: ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Point
of ( ( ) ( non
empty
)
set
) )
`1
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) )
<=
E-bound
C
: ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of ) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) ) &
p
: ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Point
of ( ( ) ( non
empty
)
set
) )
`1
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) )
>=
W-bound
C
: ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of ) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) ) &
p
: ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Point
of ( ( ) ( non
empty
)
set
) )
`2
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) )
=
S-bound
C
: ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of ) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) ) )
}
;
theorem
:: SPRECT_1:25
for
C
being ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of ) holds
LSeg
(
(
NW-corner
C
: ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of )
)
: ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) ,
(
NE-corner
C
: ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of )
)
: ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) ( non
empty
compact
V258
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) ) : ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) ) )
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) )
=
{
p
: ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Point
of ( ( ) ( non
empty
)
set
) ) where
p
is ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Point
of ( ( ) ( non
empty
)
set
) ) : (
p
: ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Point
of ( ( ) ( non
empty
)
set
) )
`1
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) )
<=
E-bound
C
: ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of ) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) ) &
p
: ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Point
of ( ( ) ( non
empty
)
set
) )
`1
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) )
>=
W-bound
C
: ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of ) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) ) &
p
: ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Point
of ( ( ) ( non
empty
)
set
) )
`2
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) )
=
N-bound
C
: ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of ) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) ) )
}
;
theorem
:: SPRECT_1:26
for
C
being ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of ) holds
LSeg
(
(
SW-corner
C
: ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of )
)
: ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) ,
(
NW-corner
C
: ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of )
)
: ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) ( non
empty
compact
V258
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) ) : ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) ) )
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) )
=
{
p
: ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Point
of ( ( ) ( non
empty
)
set
) ) where
p
is ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Point
of ( ( ) ( non
empty
)
set
) ) : (
p
: ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Point
of ( ( ) ( non
empty
)
set
) )
`1
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) )
=
W-bound
C
: ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of ) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) ) &
p
: ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Point
of ( ( ) ( non
empty
)
set
) )
`2
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) )
<=
N-bound
C
: ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of ) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) ) &
p
: ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Point
of ( ( ) ( non
empty
)
set
) )
`2
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) )
>=
S-bound
C
: ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of ) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) ) )
}
;
theorem
:: SPRECT_1:27
for
C
being ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of ) holds
(
LSeg
(
(
SW-corner
C
: ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of )
)
: ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) ,
(
NW-corner
C
: ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of )
)
: ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) ( non
empty
compact
V258
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) ) : ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) ) )
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) )
/\
(
LSeg
(
(
NW-corner
C
: ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of )
)
: ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) ,
(
NE-corner
C
: ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of )
)
: ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) ( non
empty
compact
V258
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) ) : ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) ) )
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) (
compact
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) )
=
{
(
NW-corner
C
: ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of )
)
: ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
}
: ( ( ) ( non
empty
trivial
)
set
) ;
theorem
:: SPRECT_1:28
for
C
being ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of ) holds
(
LSeg
(
(
NW-corner
C
: ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of )
)
: ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) ,
(
NE-corner
C
: ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of )
)
: ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) ( non
empty
compact
V258
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) ) : ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) ) )
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) )
/\
(
LSeg
(
(
NE-corner
C
: ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of )
)
: ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) ,
(
SE-corner
C
: ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of )
)
: ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) ( non
empty
compact
V258
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) ) : ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) ) )
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) (
compact
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) )
=
{
(
NE-corner
C
: ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of )
)
: ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
}
: ( ( ) ( non
empty
trivial
)
set
) ;
theorem
:: SPRECT_1:29
for
C
being ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of ) holds
(
LSeg
(
(
SE-corner
C
: ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of )
)
: ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) ,
(
NE-corner
C
: ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of )
)
: ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) ( non
empty
compact
V258
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) ) : ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) ) )
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) )
/\
(
LSeg
(
(
SW-corner
C
: ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of )
)
: ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) ,
(
SE-corner
C
: ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of )
)
: ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) ( non
empty
compact
V258
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) ) : ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) ) )
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) (
compact
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) )
=
{
(
SE-corner
C
: ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of )
)
: ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
}
: ( ( ) ( non
empty
trivial
)
set
) ;
theorem
:: SPRECT_1:30
for
C
being ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of ) holds
(
LSeg
(
(
NW-corner
C
: ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of )
)
: ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) ,
(
SW-corner
C
: ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of )
)
: ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) ( non
empty
compact
V258
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) ) : ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) ) )
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) )
/\
(
LSeg
(
(
SW-corner
C
: ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of )
)
: ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) ,
(
SE-corner
C
: ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of )
)
: ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) ( non
empty
compact
V258
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) ) : ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) ) )
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) (
compact
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) )
=
{
(
SW-corner
C
: ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of )
)
: ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
}
: ( ( ) ( non
empty
trivial
)
set
) ;
begin
theorem
:: SPRECT_1:31
for
D1
being ( ( non
empty
compact
non
vertical
) ( non
empty
compact
non
vertical
)
Subset
of ) holds
W-bound
D1
: ( ( non
empty
compact
non
vertical
) ( non
empty
compact
non
vertical
)
Subset
of ) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) )
<
E-bound
D1
: ( ( non
empty
compact
non
vertical
) ( non
empty
compact
non
vertical
)
Subset
of ) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) ) ;
theorem
:: SPRECT_1:32
for
D2
being ( ( non
empty
compact
non
horizontal
) ( non
empty
compact
non
horizontal
)
Subset
of ) holds
S-bound
D2
: ( ( non
empty
compact
non
horizontal
) ( non
empty
compact
non
horizontal
)
Subset
of ) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) )
<
N-bound
D2
: ( ( non
empty
compact
non
horizontal
) ( non
empty
compact
non
horizontal
)
Subset
of ) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) ) ;
theorem
:: SPRECT_1:33
for
D1
being ( ( non
empty
compact
non
vertical
) ( non
empty
compact
non
vertical
)
Subset
of ) holds
LSeg
(
(
SW-corner
D1
: ( ( non
empty
compact
non
vertical
) ( non
empty
compact
non
vertical
)
Subset
of )
)
: ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) ,
(
NW-corner
D1
: ( ( non
empty
compact
non
vertical
) ( non
empty
compact
non
vertical
)
Subset
of )
)
: ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) ( non
empty
compact
V258
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) ) : ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) ) )
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) )
misses
LSeg
(
(
SE-corner
D1
: ( ( non
empty
compact
non
vertical
) ( non
empty
compact
non
vertical
)
Subset
of )
)
: ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) ,
(
NE-corner
D1
: ( ( non
empty
compact
non
vertical
) ( non
empty
compact
non
vertical
)
Subset
of )
)
: ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) ( non
empty
compact
V258
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) ) : ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) ) )
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) ) ;
theorem
:: SPRECT_1:34
for
D2
being ( ( non
empty
compact
non
horizontal
) ( non
empty
compact
non
horizontal
)
Subset
of ) holds
LSeg
(
(
SW-corner
D2
: ( ( non
empty
compact
non
horizontal
) ( non
empty
compact
non
horizontal
)
Subset
of )
)
: ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) ,
(
SE-corner
D2
: ( ( non
empty
compact
non
horizontal
) ( non
empty
compact
non
horizontal
)
Subset
of )
)
: ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) ( non
empty
compact
V258
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) ) : ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) ) )
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) )
misses
LSeg
(
(
NW-corner
D2
: ( ( non
empty
compact
non
horizontal
) ( non
empty
compact
non
horizontal
)
Subset
of )
)
: ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) ,
(
NE-corner
D2
: ( ( non
empty
compact
non
horizontal
) ( non
empty
compact
non
horizontal
)
Subset
of )
)
: ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) ( non
empty
compact
V258
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) ) : ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) ) )
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) ) ;
begin
definition
let
C
be ( ( ) ( )
Subset
of ) ;
func
SpStSeq
C
->
( ( ) (
V19
()
V22
(
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
V23
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V32
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of ( ( ) ( non
empty
)
set
) )
equals
:: SPRECT_1:def 1
<*
(
NW-corner
C
: ( ( ) ( )
set
)
)
: ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) ,
(
NE-corner
C
: ( ( ) ( )
set
)
)
: ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) ,
(
SE-corner
C
: ( ( ) ( )
set
)
)
: ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
*>
: ( ( ) ( non
empty
non
trivial
V19
()
V22
(
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
V23
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V32
() 3 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
FinSubsequence-like
)
FinSequence
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
^
<*
(
SW-corner
C
: ( ( ) ( )
set
)
)
: ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) ,
(
NW-corner
C
: ( ( ) ( )
set
)
)
: ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
*>
: ( ( ) ( non
empty
non
trivial
V19
()
V22
(
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
V23
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V32
() 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
FinSubsequence-like
)
FinSequence
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( non
empty
V19
()
V22
(
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
V23
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V32
() 3 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
+
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
FinSubsequence-like
)
FinSequence
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) ;
end;
theorem
:: SPRECT_1:35
for
S
being ( ( ) ( )
Subset
of ) holds
(
SpStSeq
S
: ( ( ) ( )
Subset
of )
)
: ( ( ) (
V19
()
V22
(
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
V23
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V32
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of ( ( ) ( non
empty
)
set
) )
/.
1 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
=
NW-corner
S
: ( ( ) ( )
Subset
of ) : ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) ;
theorem
:: SPRECT_1:36
for
S
being ( ( ) ( )
Subset
of ) holds
(
SpStSeq
S
: ( ( ) ( )
Subset
of )
)
: ( ( ) (
V19
()
V22
(
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
V23
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V32
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of ( ( ) ( non
empty
)
set
) )
/.
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
=
NE-corner
S
: ( ( ) ( )
Subset
of ) : ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) ;
theorem
:: SPRECT_1:37
for
S
being ( ( ) ( )
Subset
of ) holds
(
SpStSeq
S
: ( ( ) ( )
Subset
of )
)
: ( ( ) (
V19
()
V22
(
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
V23
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V32
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of ( ( ) ( non
empty
)
set
) )
/.
3 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
=
SE-corner
S
: ( ( ) ( )
Subset
of ) : ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) ;
theorem
:: SPRECT_1:38
for
S
being ( ( ) ( )
Subset
of ) holds
(
SpStSeq
S
: ( ( ) ( )
Subset
of )
)
: ( ( ) (
V19
()
V22
(
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
V23
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V32
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of ( ( ) ( non
empty
)
set
) )
/.
4 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
=
SW-corner
S
: ( ( ) ( )
Subset
of ) : ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) ;
theorem
:: SPRECT_1:39
for
S
being ( ( ) ( )
Subset
of ) holds
(
SpStSeq
S
: ( ( ) ( )
Subset
of )
)
: ( ( ) (
V19
()
V22
(
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
V23
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V32
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of ( ( ) ( non
empty
)
set
) )
/.
5 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
=
NW-corner
S
: ( ( ) ( )
Subset
of ) : ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) ;
theorem
:: SPRECT_1:40
for
S
being ( ( ) ( )
Subset
of ) holds
len
(
SpStSeq
S
: ( ( ) ( )
Subset
of )
)
: ( ( ) (
V19
()
V22
(
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
V23
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V32
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of ( ( ) ( non
empty
)
set
) ) : ( ( ) (
ordinal
natural
V11
()
real
ext-real
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
=
5 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) ) ;
theorem
:: SPRECT_1:41
for
S
being ( ( ) ( )
Subset
of ) holds
L~
(
SpStSeq
S
: ( ( ) ( )
Subset
of )
)
: ( ( ) (
V19
()
V22
(
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
V23
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V32
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of ( ( ) ( non
empty
)
set
) ) : ( ( ) (
compact
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) )
=
(
(
LSeg
(
(
NW-corner
S
: ( ( ) ( )
Subset
of )
)
: ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) ,
(
NE-corner
S
: ( ( ) ( )
Subset
of )
)
: ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) ( non
empty
compact
V258
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) ) : ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) ) )
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) )
\/
(
LSeg
(
(
NE-corner
S
: ( ( ) ( )
Subset
of )
)
: ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) ,
(
SE-corner
S
: ( ( ) ( )
Subset
of )
)
: ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) ( non
empty
compact
V258
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) ) : ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) ) )
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) )
)
: ( ( ) ( non
empty
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) )
\/
(
(
LSeg
(
(
SE-corner
S
: ( ( ) ( )
Subset
of )
)
: ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) ,
(
SW-corner
S
: ( ( ) ( )
Subset
of )
)
: ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) ( non
empty
compact
V258
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) ) : ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) ) )
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) )
\/
(
LSeg
(
(
SW-corner
S
: ( ( ) ( )
Subset
of )
)
: ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) ,
(
NW-corner
S
: ( ( ) ( )
Subset
of )
)
: ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) ( non
empty
compact
V258
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) ) : ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) ) )
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) )
)
: ( ( ) ( non
empty
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( non
empty
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) ) ;
registration
let
D
be ( ( non
empty
compact
non
vertical
) ( non
empty
compact
non
vertical
)
Subset
of ) ;
cluster
SpStSeq
D
: ( ( non
empty
compact
non
vertical
) ( non
empty
compact
non
vertical
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) (
V19
()
V22
(
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
V23
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V32
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of ( ( ) ( non
empty
)
set
) )
->
non
constant
;
end;
registration
let
D
be ( ( non
empty
compact
non
horizontal
) ( non
empty
compact
non
horizontal
)
Subset
of ) ;
cluster
SpStSeq
D
: ( ( non
empty
compact
non
horizontal
) ( non
empty
compact
non
horizontal
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) (
V19
()
V22
(
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
V23
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V32
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of ( ( ) ( non
empty
)
set
) )
->
non
constant
;
end;
registration
let
D
be ( ( non
empty
compact
non
horizontal
non
vertical
) ( non
empty
compact
non
horizontal
non
vertical
)
Subset
of ) ;
cluster
SpStSeq
D
: ( ( 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
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( non
empty
non
trivial
V19
()
V22
(
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
V23
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V32
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of ( ( ) ( non
empty
)
set
) )
->
circular
special
unfolded
s.c.c.
standard
;
end;
theorem
:: SPRECT_1:42
for
D
being ( ( non
empty
compact
non
horizontal
non
vertical
) ( non
empty
compact
non
horizontal
non
vertical
)
Subset
of ) holds
L~
(
SpStSeq
D
: ( ( non
empty
compact
non
horizontal
non
vertical
) ( non
empty
compact
non
horizontal
non
vertical
)
Subset
of )
)
: ( ( ) ( non
empty
non
trivial
V19
()
V22
(
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
V23
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V32
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
)
FinSequence
of ( ( ) ( non
empty
)
set
) ) : ( ( ) ( non
empty
compact
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) )
=
[.
(
W-bound
D
: ( ( non
empty
compact
non
horizontal
non
vertical
) ( non
empty
compact
non
horizontal
non
vertical
)
Subset
of )
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) ) ,
(
E-bound
D
: ( ( non
empty
compact
non
horizontal
non
vertical
) ( non
empty
compact
non
horizontal
non
vertical
)
Subset
of )
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) ) ,
(
S-bound
D
: ( ( non
empty
compact
non
horizontal
non
vertical
) ( non
empty
compact
non
horizontal
non
vertical
)
Subset
of )
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) ) ,
(
N-bound
D
: ( ( non
empty
compact
non
horizontal
non
vertical
) ( non
empty
compact
non
horizontal
non
vertical
)
Subset
of )
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) )
.]
: ( ( ) ( non
empty
compact
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) ) ;
registration
let
T
be ( ( non
empty
TopSpace-like
) ( non
empty
TopSpace-like
)
TopSpace
) ;
let
X
be ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of ) ;
let
f
be ( (
Function-like
V44
( the
carrier
of
T
: ( ( non
empty
TopSpace-like
) ( non
empty
TopSpace-like
)
TopSpace
) : ( ( ) ( non
empty
)
set
) ,
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) )
continuous
) (
V19
()
V22
( the
carrier
of
T
: ( ( non
empty
TopSpace-like
) ( non
empty
TopSpace-like
)
TopSpace
) : ( ( ) ( non
empty
)
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) )
Function-like
V44
( the
carrier
of
T
: ( ( non
empty
TopSpace-like
) ( non
empty
TopSpace-like
)
TopSpace
) : ( ( ) ( non
empty
)
set
) ,
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) )
V189
()
V190
()
V191
()
continuous
)
RealMap
of ( ( ) ( non
empty
)
set
) ) ;
cluster
f
: ( (
Function-like
V44
( the
carrier
of
T
: ( ( non
empty
TopSpace-like
) ( non
empty
TopSpace-like
)
TopStruct
) : ( ( ) ( non
empty
)
set
) ,
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) )
continuous
) (
V19
()
V22
( the
carrier
of
T
: ( ( non
empty
TopSpace-like
) ( non
empty
TopSpace-like
)
TopStruct
) : ( ( ) ( non
empty
)
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) )
Function-like
V44
( the
carrier
of
T
: ( ( non
empty
TopSpace-like
) ( non
empty
TopSpace-like
)
TopStruct
) : ( ( ) ( non
empty
)
set
) ,
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) )
V189
()
V190
()
V191
()
continuous
)
Element
of
bool
[:
the
carrier
of
T
: ( ( non
empty
TopSpace-like
) ( non
empty
TopSpace-like
)
TopStruct
) : ( ( ) ( non
empty
)
set
) ,
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
)
:]
: ( ( ) ( non
empty
V189
()
V190
()
V191
() )
set
) : ( ( ) ( non
empty
)
set
) )
.:
X
: ( ( non
empty
compact
) ( non
empty
compact
)
Element
of
bool
the
carrier
of
T
: ( ( non
empty
TopSpace-like
) ( non
empty
TopSpace-like
)
TopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( non
empty
V199
()
V200
()
V201
() )
set
)
->
bounded_below
;
cluster
f
: ( (
Function-like
V44
( the
carrier
of
T
: ( ( non
empty
TopSpace-like
) ( non
empty
TopSpace-like
)
TopStruct
) : ( ( ) ( non
empty
)
set
) ,
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) )
continuous
) (
V19
()
V22
( the
carrier
of
T
: ( ( non
empty
TopSpace-like
) ( non
empty
TopSpace-like
)
TopStruct
) : ( ( ) ( non
empty
)
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) )
Function-like
V44
( the
carrier
of
T
: ( ( non
empty
TopSpace-like
) ( non
empty
TopSpace-like
)
TopStruct
) : ( ( ) ( non
empty
)
set
) ,
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) )
V189
()
V190
()
V191
()
continuous
)
Element
of
bool
[:
the
carrier
of
T
: ( ( non
empty
TopSpace-like
) ( non
empty
TopSpace-like
)
TopStruct
) : ( ( ) ( non
empty
)
set
) ,
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
)
:]
: ( ( ) ( non
empty
V189
()
V190
()
V191
() )
set
) : ( ( ) ( non
empty
)
set
) )
.:
X
: ( ( non
empty
compact
) ( non
empty
compact
)
Element
of
bool
the
carrier
of
T
: ( ( non
empty
TopSpace-like
) ( non
empty
TopSpace-like
)
TopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( non
empty
V199
()
V200
()
V201
() )
set
)
->
bounded_above
;
end;
theorem
:: SPRECT_1:43
for
S
being ( ( ) ( )
Subset
of ) holds
W-bound
S
: ( ( ) ( )
Subset
of ) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) )
=
lower_bound
(
proj1
: ( (
Function-like
V44
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ,
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) ) ) (
V19
()
V22
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) )
Function-like
V44
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ,
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) )
V189
()
V190
()
V191
()
continuous
)
Element
of
bool
[:
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ,
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
)
:]
: ( ( ) ( non
empty
V189
()
V190
()
V191
() )
set
) : ( ( ) ( non
empty
)
set
) )
.:
S
: ( ( ) ( )
Subset
of )
)
: ( ( ) (
V199
()
V200
()
V201
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) ) ;
theorem
:: SPRECT_1:44
for
S
being ( ( ) ( )
Subset
of ) holds
S-bound
S
: ( ( ) ( )
Subset
of ) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) )
=
lower_bound
(
proj2
: ( (
Function-like
V44
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ,
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) ) ) (
V19
()
V22
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) )
Function-like
V44
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ,
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) )
V189
()
V190
()
V191
()
continuous
)
Element
of
bool
[:
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ,
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
)
:]
: ( ( ) ( non
empty
V189
()
V190
()
V191
() )
set
) : ( ( ) ( non
empty
)
set
) )
.:
S
: ( ( ) ( )
Subset
of )
)
: ( ( ) (
V199
()
V200
()
V201
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) ) ;
theorem
:: SPRECT_1:45
for
S
being ( ( ) ( )
Subset
of ) holds
N-bound
S
: ( ( ) ( )
Subset
of ) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) )
=
upper_bound
(
proj2
: ( (
Function-like
V44
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ,
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) ) ) (
V19
()
V22
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) )
Function-like
V44
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ,
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) )
V189
()
V190
()
V191
()
continuous
)
Element
of
bool
[:
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ,
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
)
:]
: ( ( ) ( non
empty
V189
()
V190
()
V191
() )
set
) : ( ( ) ( non
empty
)
set
) )
.:
S
: ( ( ) ( )
Subset
of )
)
: ( ( ) (
V199
()
V200
()
V201
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) ) ;
theorem
:: SPRECT_1:46
for
S
being ( ( ) ( )
Subset
of ) holds
E-bound
S
: ( ( ) ( )
Subset
of ) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) )
=
upper_bound
(
proj1
: ( (
Function-like
V44
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ,
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) ) ) (
V19
()
V22
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) )
Function-like
V44
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ,
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) )
V189
()
V190
()
V191
()
continuous
)
Element
of
bool
[:
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ,
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
)
:]
: ( ( ) ( non
empty
V189
()
V190
()
V191
() )
set
) : ( ( ) ( non
empty
)
set
) )
.:
S
: ( ( ) ( )
Subset
of )
)
: ( ( ) (
V199
()
V200
()
V201
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) ) ;
theorem
:: SPRECT_1:47
for
S
being ( ( ) ( )
Subset
of )
for
C1
,
C2
being ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of ) st
S
: ( ( ) ( )
Subset
of )
=
C1
: ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of )
\/
C2
: ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of ) : ( ( ) ( non
empty
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) ) holds
W-bound
S
: ( ( ) ( )
Subset
of ) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) )
=
min
(
(
W-bound
C1
: ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of )
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) ) ,
(
W-bound
C2
: ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of )
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) ) ) : ( ( ) (
V11
()
real
ext-real
)
set
) ;
theorem
:: SPRECT_1:48
for
S
being ( ( ) ( )
Subset
of )
for
C1
,
C2
being ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of ) st
S
: ( ( ) ( )
Subset
of )
=
C1
: ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of )
\/
C2
: ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of ) : ( ( ) ( non
empty
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) ) holds
S-bound
S
: ( ( ) ( )
Subset
of ) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) )
=
min
(
(
S-bound
C1
: ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of )
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) ) ,
(
S-bound
C2
: ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of )
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) ) ) : ( ( ) (
V11
()
real
ext-real
)
set
) ;
theorem
:: SPRECT_1:49
for
S
being ( ( ) ( )
Subset
of )
for
C1
,
C2
being ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of ) st
S
: ( ( ) ( )
Subset
of )
=
C1
: ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of )
\/
C2
: ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of ) : ( ( ) ( non
empty
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) ) holds
N-bound
S
: ( ( ) ( )
Subset
of ) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) )
=
max
(
(
N-bound
C1
: ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of )
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) ) ,
(
N-bound
C2
: ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of )
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) ) ) : ( ( ) (
V11
()
real
ext-real
)
set
) ;
theorem
:: SPRECT_1:50
for
S
being ( ( ) ( )
Subset
of )
for
C1
,
C2
being ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of ) st
S
: ( ( ) ( )
Subset
of )
=
C1
: ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of )
\/
C2
: ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of ) : ( ( ) ( non
empty
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) ) holds
E-bound
S
: ( ( ) ( )
Subset
of ) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) )
=
max
(
(
E-bound
C1
: ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of )
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) ) ,
(
E-bound
C2
: ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of )
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) ) ) : ( ( ) (
V11
()
real
ext-real
)
set
) ;
registration
let
r1
,
r2
be ( (
real
) (
V11
()
real
ext-real
)
number
) ;
cluster
K76
(
r1
: ( (
real
) (
V11
()
real
ext-real
)
set
) ,
r2
: ( (
real
) (
V11
()
real
ext-real
)
set
) ) : ( ( ) (
V245
() )
set
)
->
real-bounded
for ( ( ) (
V199
()
V200
()
V201
() )
Subset
of ( ( ) ( non
empty
)
set
) ) ;
end;
theorem
:: SPRECT_1:51
for
r1
,
r2
,
t
being ( ( ) (
V11
()
real
ext-real
)
Real
) st
r1
: ( ( ) (
V11
()
real
ext-real
)
Real
)
<=
r2
: ( ( ) (
V11
()
real
ext-real
)
Real
) holds
(
t
: ( ( ) (
V11
()
real
ext-real
)
Real
)
in
[.
r1
: ( ( ) (
V11
()
real
ext-real
)
Real
) ,
r2
: ( ( ) (
V11
()
real
ext-real
)
Real
)
.]
: ( ( ) (
V199
()
V200
()
V201
()
bounded_below
bounded_above
real-bounded
V245
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) iff ex
s1
being ( ( ) (
V11
()
real
ext-real
)
Real
) st
(
0
: ( ( ) (
empty
trivial
ordinal
natural
V11
()
real
ext-real
non
positive
non
negative
Function-like
functional
FinSequence-membered
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
V245
() )
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
<=
s1
: ( ( ) (
V11
()
real
ext-real
)
Real
) &
s1
: ( ( ) (
V11
()
real
ext-real
)
Real
)
<=
1 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) ) &
t
: ( ( ) (
V11
()
real
ext-real
)
Real
)
=
(
s1
: ( ( ) (
V11
()
real
ext-real
)
Real
)
*
r1
: ( ( ) (
V11
()
real
ext-real
)
Real
)
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) )
+
(
(
1 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-
s1
: ( ( ) (
V11
()
real
ext-real
)
Real
)
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) )
*
r2
: ( ( ) (
V11
()
real
ext-real
)
Real
)
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) ) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) ) ) ) ;
theorem
:: SPRECT_1:52
for
p
,
q
being ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Point
of ( ( ) ( non
empty
)
set
) ) st
p
: ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Point
of ( ( ) ( non
empty
)
set
) )
`1
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) )
<=
q
: ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Point
of ( ( ) ( non
empty
)
set
) )
`1
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) ) holds
proj1
: ( (
Function-like
V44
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ,
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) ) ) (
V19
()
V22
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) )
Function-like
V44
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ,
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) )
V189
()
V190
()
V191
()
continuous
)
Element
of
bool
[:
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ,
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
)
:]
: ( ( ) ( non
empty
V189
()
V190
()
V191
() )
set
) : ( ( ) ( non
empty
)
set
) )
.:
(
LSeg
(
p
: ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Point
of ( ( ) ( non
empty
)
set
) ) ,
q
: ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Point
of ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) ( non
empty
compact
V258
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) ) : ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) ) )
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( non
empty
V199
()
V200
()
V201
()
bounded_below
bounded_above
real-bounded
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) )
=
[.
(
p
: ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Point
of ( ( ) ( non
empty
)
set
) )
`1
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) ) ,
(
q
: ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Point
of ( ( ) ( non
empty
)
set
) )
`1
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) )
.]
: ( ( ) (
V199
()
V200
()
V201
()
bounded_below
bounded_above
real-bounded
V245
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) ;
theorem
:: SPRECT_1:53
for
p
,
q
being ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Point
of ( ( ) ( non
empty
)
set
) ) st
p
: ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Point
of ( ( ) ( non
empty
)
set
) )
`2
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) )
<=
q
: ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Point
of ( ( ) ( non
empty
)
set
) )
`2
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) ) holds
proj2
: ( (
Function-like
V44
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ,
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) ) ) (
V19
()
V22
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) )
Function-like
V44
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ,
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) )
V189
()
V190
()
V191
()
continuous
)
Element
of
bool
[:
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ,
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
)
:]
: ( ( ) ( non
empty
V189
()
V190
()
V191
() )
set
) : ( ( ) ( non
empty
)
set
) )
.:
(
LSeg
(
p
: ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Point
of ( ( ) ( non
empty
)
set
) ) ,
q
: ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Point
of ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) ( non
empty
compact
V258
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) ) : ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) ) )
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( non
empty
V199
()
V200
()
V201
()
bounded_below
bounded_above
real-bounded
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) )
=
[.
(
p
: ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Point
of ( ( ) ( non
empty
)
set
) )
`2
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) ) ,
(
q
: ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Point
of ( ( ) ( non
empty
)
set
) )
`2
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) )
.]
: ( ( ) (
V199
()
V200
()
V201
()
bounded_below
bounded_above
real-bounded
V245
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) ;
theorem
:: SPRECT_1:54
for
p
,
q
being ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Point
of ( ( ) ( non
empty
)
set
) ) st
p
: ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Point
of ( ( ) ( non
empty
)
set
) )
`1
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) )
<=
q
: ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Point
of ( ( ) ( non
empty
)
set
) )
`1
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) ) holds
W-bound
(
LSeg
(
p
: ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Point
of ( ( ) ( non
empty
)
set
) ) ,
q
: ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Point
of ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) ( non
empty
compact
V258
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) ) : ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) ) )
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) )
=
p
: ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Point
of ( ( ) ( non
empty
)
set
) )
`1
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) ) ;
theorem
:: SPRECT_1:55
for
p
,
q
being ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Point
of ( ( ) ( non
empty
)
set
) ) st
p
: ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Point
of ( ( ) ( non
empty
)
set
) )
`2
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) )
<=
q
: ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Point
of ( ( ) ( non
empty
)
set
) )
`2
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) ) holds
S-bound
(
LSeg
(
p
: ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Point
of ( ( ) ( non
empty
)
set
) ) ,
q
: ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Point
of ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) ( non
empty
compact
V258
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) ) : ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) ) )
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) )
=
p
: ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Point
of ( ( ) ( non
empty
)
set
) )
`2
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) ) ;
theorem
:: SPRECT_1:56
for
p
,
q
being ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Point
of ( ( ) ( non
empty
)
set
) ) st
p
: ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Point
of ( ( ) ( non
empty
)
set
) )
`2
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) )
<=
q
: ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Point
of ( ( ) ( non
empty
)
set
) )
`2
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) ) holds
N-bound
(
LSeg
(
p
: ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Point
of ( ( ) ( non
empty
)
set
) ) ,
q
: ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Point
of ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) ( non
empty
compact
V258
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) ) : ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) ) )
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) )
=
q
: ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Point
of ( ( ) ( non
empty
)
set
) )
`2
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) ) ;
theorem
:: SPRECT_1:57
for
p
,
q
being ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Point
of ( ( ) ( non
empty
)
set
) ) st
p
: ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Point
of ( ( ) ( non
empty
)
set
) )
`1
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) )
<=
q
: ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Point
of ( ( ) ( non
empty
)
set
) )
`1
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) ) holds
E-bound
(
LSeg
(
p
: ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Point
of ( ( ) ( non
empty
)
set
) ) ,
q
: ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Point
of ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) ( non
empty
compact
V258
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) ) : ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) ) )
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) )
=
q
: ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Point
of ( ( ) ( non
empty
)
set
) )
`1
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) ) ;
theorem
:: SPRECT_1:58
for
C
being ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of ) holds
W-bound
(
L~
(
SpStSeq
C
: ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of )
)
: ( ( ) (
V19
()
V22
(
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
V23
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V32
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of ( ( ) ( non
empty
)
set
) )
)
: ( ( ) (
compact
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) )
=
W-bound
C
: ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of ) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) ) ;
theorem
:: SPRECT_1:59
for
C
being ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of ) holds
S-bound
(
L~
(
SpStSeq
C
: ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of )
)
: ( ( ) (
V19
()
V22
(
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
V23
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V32
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of ( ( ) ( non
empty
)
set
) )
)
: ( ( ) (
compact
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) )
=
S-bound
C
: ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of ) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) ) ;
theorem
:: SPRECT_1:60
for
C
being ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of ) holds
N-bound
(
L~
(
SpStSeq
C
: ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of )
)
: ( ( ) (
V19
()
V22
(
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
V23
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V32
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of ( ( ) ( non
empty
)
set
) )
)
: ( ( ) (
compact
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) )
=
N-bound
C
: ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of ) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) ) ;
theorem
:: SPRECT_1:61
for
C
being ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of ) holds
E-bound
(
L~
(
SpStSeq
C
: ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of )
)
: ( ( ) (
V19
()
V22
(
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
V23
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V32
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of ( ( ) ( non
empty
)
set
) )
)
: ( ( ) (
compact
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) )
=
E-bound
C
: ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of ) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) ) ;
theorem
:: SPRECT_1:62
for
C
being ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of ) holds
NW-corner
(
L~
(
SpStSeq
C
: ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of )
)
: ( ( ) (
V19
()
V22
(
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
V23
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V32
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of ( ( ) ( non
empty
)
set
) )
)
: ( ( ) (
compact
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
=
NW-corner
C
: ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of ) : ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) ;
theorem
:: SPRECT_1:63
for
C
being ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of ) holds
NE-corner
(
L~
(
SpStSeq
C
: ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of )
)
: ( ( ) (
V19
()
V22
(
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
V23
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V32
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of ( ( ) ( non
empty
)
set
) )
)
: ( ( ) (
compact
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
=
NE-corner
C
: ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of ) : ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) ;
theorem
:: SPRECT_1:64
for
C
being ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of ) holds
SW-corner
(
L~
(
SpStSeq
C
: ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of )
)
: ( ( ) (
V19
()
V22
(
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
V23
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V32
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of ( ( ) ( non
empty
)
set
) )
)
: ( ( ) (
compact
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
=
SW-corner
C
: ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of ) : ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) ;
theorem
:: SPRECT_1:65
for
C
being ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of ) holds
SE-corner
(
L~
(
SpStSeq
C
: ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of )
)
: ( ( ) (
V19
()
V22
(
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
V23
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V32
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of ( ( ) ( non
empty
)
set
) )
)
: ( ( ) (
compact
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
=
SE-corner
C
: ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of ) : ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) ;
theorem
:: SPRECT_1:66
for
C
being ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of ) holds
W-most
(
L~
(
SpStSeq
C
: ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of )
)
: ( ( ) (
V19
()
V22
(
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
V23
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V32
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of ( ( ) ( non
empty
)
set
) )
)
: ( ( ) (
compact
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( )
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) )
=
LSeg
(
(
SW-corner
C
: ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of )
)
: ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) ,
(
NW-corner
C
: ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of )
)
: ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) ( non
empty
compact
V258
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) ) : ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) ) )
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) ) ;
theorem
:: SPRECT_1:67
for
C
being ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of ) holds
N-most
(
L~
(
SpStSeq
C
: ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of )
)
: ( ( ) (
V19
()
V22
(
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
V23
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V32
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of ( ( ) ( non
empty
)
set
) )
)
: ( ( ) (
compact
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( )
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) )
=
LSeg
(
(
NW-corner
C
: ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of )
)
: ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) ,
(
NE-corner
C
: ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of )
)
: ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) ( non
empty
compact
V258
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) ) : ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) ) )
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) ) ;
theorem
:: SPRECT_1:68
for
C
being ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of ) holds
S-most
(
L~
(
SpStSeq
C
: ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of )
)
: ( ( ) (
V19
()
V22
(
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
V23
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V32
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of ( ( ) ( non
empty
)
set
) )
)
: ( ( ) (
compact
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( )
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) )
=
LSeg
(
(
SW-corner
C
: ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of )
)
: ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) ,
(
SE-corner
C
: ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of )
)
: ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) ( non
empty
compact
V258
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) ) : ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) ) )
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) ) ;
theorem
:: SPRECT_1:69
for
C
being ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of ) holds
E-most
(
L~
(
SpStSeq
C
: ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of )
)
: ( ( ) (
V19
()
V22
(
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
V23
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V32
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of ( ( ) ( non
empty
)
set
) )
)
: ( ( ) (
compact
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( )
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) )
=
LSeg
(
(
SE-corner
C
: ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of )
)
: ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) ,
(
NE-corner
C
: ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of )
)
: ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) ( non
empty
compact
V258
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) ) : ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) ) )
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) ) ;
theorem
:: SPRECT_1:70
for
C
being ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of ) holds
proj2
: ( (
Function-like
V44
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ,
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) ) ) (
V19
()
V22
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) )
Function-like
V44
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ,
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) )
V189
()
V190
()
V191
()
continuous
)
Element
of
bool
[:
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ,
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
)
:]
: ( ( ) ( non
empty
V189
()
V190
()
V191
() )
set
) : ( ( ) ( non
empty
)
set
) )
.:
(
LSeg
(
(
SW-corner
C
: ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of )
)
: ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) ,
(
NW-corner
C
: ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of )
)
: ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) ( non
empty
compact
V258
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) ) : ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) ) )
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( non
empty
V199
()
V200
()
V201
()
bounded_below
bounded_above
real-bounded
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) )
=
[.
(
S-bound
C
: ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of )
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) ) ,
(
N-bound
C
: ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of )
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) )
.]
: ( ( ) (
V199
()
V200
()
V201
()
bounded_below
bounded_above
real-bounded
V245
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) ;
theorem
:: SPRECT_1:71
for
C
being ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of ) holds
proj1
: ( (
Function-like
V44
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ,
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) ) ) (
V19
()
V22
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) )
Function-like
V44
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ,
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) )
V189
()
V190
()
V191
()
continuous
)
Element
of
bool
[:
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ,
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
)
:]
: ( ( ) ( non
empty
V189
()
V190
()
V191
() )
set
) : ( ( ) ( non
empty
)
set
) )
.:
(
LSeg
(
(
NW-corner
C
: ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of )
)
: ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) ,
(
NE-corner
C
: ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of )
)
: ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) ( non
empty
compact
V258
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) ) : ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) ) )
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( non
empty
V199
()
V200
()
V201
()
bounded_below
bounded_above
real-bounded
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) )
=
[.
(
W-bound
C
: ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of )
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) ) ,
(
E-bound
C
: ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of )
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) )
.]
: ( ( ) (
V199
()
V200
()
V201
()
bounded_below
bounded_above
real-bounded
V245
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) ;
theorem
:: SPRECT_1:72
for
C
being ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of ) holds
proj2
: ( (
Function-like
V44
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ,
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) ) ) (
V19
()
V22
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) )
Function-like
V44
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ,
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) )
V189
()
V190
()
V191
()
continuous
)
Element
of
bool
[:
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ,
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
)
:]
: ( ( ) ( non
empty
V189
()
V190
()
V191
() )
set
) : ( ( ) ( non
empty
)
set
) )
.:
(
LSeg
(
(
NE-corner
C
: ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of )
)
: ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) ,
(
SE-corner
C
: ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of )
)
: ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) ( non
empty
compact
V258
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) ) : ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) ) )
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( non
empty
V199
()
V200
()
V201
()
bounded_below
bounded_above
real-bounded
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) )
=
[.
(
S-bound
C
: ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of )
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) ) ,
(
N-bound
C
: ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of )
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) )
.]
: ( ( ) (
V199
()
V200
()
V201
()
bounded_below
bounded_above
real-bounded
V245
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) ;
theorem
:: SPRECT_1:73
for
C
being ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of ) holds
proj1
: ( (
Function-like
V44
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ,
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) ) ) (
V19
()
V22
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) )
Function-like
V44
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ,
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) )
V189
()
V190
()
V191
()
continuous
)
Element
of
bool
[:
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ,
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
)
:]
: ( ( ) ( non
empty
V189
()
V190
()
V191
() )
set
) : ( ( ) ( non
empty
)
set
) )
.:
(
LSeg
(
(
SE-corner
C
: ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of )
)
: ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) ,
(
SW-corner
C
: ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of )
)
: ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) ( non
empty
compact
V258
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) ) : ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) ) )
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( non
empty
V199
()
V200
()
V201
()
bounded_below
bounded_above
real-bounded
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) )
=
[.
(
W-bound
C
: ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of )
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) ) ,
(
E-bound
C
: ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of )
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) )
.]
: ( ( ) (
V199
()
V200
()
V201
()
bounded_below
bounded_above
real-bounded
V245
() )
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) ;
theorem
:: SPRECT_1:74
for
C
being ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of ) holds
W-min
(
L~
(
SpStSeq
C
: ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of )
)
: ( ( ) (
V19
()
V22
(
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
V23
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V32
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of ( ( ) ( non
empty
)
set
) )
)
: ( ( ) (
compact
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
=
SW-corner
C
: ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of ) : ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) ;
theorem
:: SPRECT_1:75
for
C
being ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of ) holds
W-max
(
L~
(
SpStSeq
C
: ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of )
)
: ( ( ) (
V19
()
V22
(
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
V23
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V32
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of ( ( ) ( non
empty
)
set
) )
)
: ( ( ) (
compact
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
=
NW-corner
C
: ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of ) : ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) ;
theorem
:: SPRECT_1:76
for
C
being ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of ) holds
N-min
(
L~
(
SpStSeq
C
: ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of )
)
: ( ( ) (
V19
()
V22
(
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
V23
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V32
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of ( ( ) ( non
empty
)
set
) )
)
: ( ( ) (
compact
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
=
NW-corner
C
: ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of ) : ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) ;
theorem
:: SPRECT_1:77
for
C
being ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of ) holds
N-max
(
L~
(
SpStSeq
C
: ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of )
)
: ( ( ) (
V19
()
V22
(
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
V23
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V32
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of ( ( ) ( non
empty
)
set
) )
)
: ( ( ) (
compact
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
=
NE-corner
C
: ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of ) : ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) ;
theorem
:: SPRECT_1:78
for
C
being ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of ) holds
E-min
(
L~
(
SpStSeq
C
: ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of )
)
: ( ( ) (
V19
()
V22
(
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
V23
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V32
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of ( ( ) ( non
empty
)
set
) )
)
: ( ( ) (
compact
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
=
SE-corner
C
: ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of ) : ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) ;
theorem
:: SPRECT_1:79
for
C
being ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of ) holds
E-max
(
L~
(
SpStSeq
C
: ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of )
)
: ( ( ) (
V19
()
V22
(
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
V23
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V32
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of ( ( ) ( non
empty
)
set
) )
)
: ( ( ) (
compact
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
=
NE-corner
C
: ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of ) : ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) ;
theorem
:: SPRECT_1:80
for
C
being ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of ) holds
S-min
(
L~
(
SpStSeq
C
: ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of )
)
: ( ( ) (
V19
()
V22
(
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
V23
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V32
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of ( ( ) ( non
empty
)
set
) )
)
: ( ( ) (
compact
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
=
SW-corner
C
: ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of ) : ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) ;
theorem
:: SPRECT_1:81
for
C
being ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of ) holds
S-max
(
L~
(
SpStSeq
C
: ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of )
)
: ( ( ) (
V19
()
V22
(
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
V23
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V32
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of ( ( ) ( non
empty
)
set
) )
)
: ( ( ) (
compact
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
=
SE-corner
C
: ( ( non
empty
compact
) ( non
empty
compact
)
Subset
of ) : ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) ;
begin
definition
let
f
be ( ( ) (
V19
()
V22
(
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
V23
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V32
()
FinSequence-like
FinSubsequence-like
)
FinSequence
of ( ( ) ( non
empty
)
set
) ) ;
attr
f
is
rectangular
means
:: SPRECT_1:def 2
ex
D
being ( ( non
empty
compact
non
horizontal
non
vertical
) ( non
empty
compact
non
horizontal
non
vertical
)
Subset
of ) st
f
: ( ( ) ( )
set
)
=
SpStSeq
D
: ( ( non
empty
compact
non
horizontal
non
vertical
) ( non
empty
compact
non
horizontal
non
vertical
)
Subset
of ) : ( ( ) ( non
empty
non
trivial
V19
()
V22
(
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
V23
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V32
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
)
FinSequence
of ( ( ) ( non
empty
)
set
) ) ;
end;
registration
let
D
be ( ( non
empty
compact
non
horizontal
non
vertical
) ( non
empty
compact
non
horizontal
non
vertical
)
Subset
of ) ;
cluster
SpStSeq
D
: ( ( 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
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( non
empty
non
trivial
V19
()
V22
(
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
V23
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V32
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
)
FinSequence
of ( ( ) ( non
empty
)
set
) )
->
rectangular
;
end;
registration
cluster
V19
()
V22
(
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
V23
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V32
()
FinSequence-like
FinSubsequence-like
rectangular
for ( ( ) ( )
FinSequence
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) ;
end;
theorem
:: SPRECT_1:82
for
s
being ( (
rectangular
) (
V19
()
V22
(
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
V23
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V32
()
FinSequence-like
FinSubsequence-like
rectangular
)
FinSequence
of ( ( ) ( non
empty
)
set
) ) holds
len
s
: ( (
rectangular
) (
V19
()
V22
(
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
V23
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V32
()
FinSequence-like
FinSubsequence-like
rectangular
)
FinSequence
of ( ( ) ( non
empty
)
set
) ) : ( ( ) (
ordinal
natural
V11
()
real
ext-real
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
=
5 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) ) ;
registration
cluster
rectangular
->
non
constant
for ( ( ) ( )
FinSequence
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) ;
end;
registration
cluster
non
empty
rectangular
->
non
empty
circular
special
unfolded
s.c.c.
standard
for ( ( ) ( )
FinSequence
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) ;
end;
theorem
:: SPRECT_1:83
for
s
being ( (
rectangular
) (
V19
()
V22
(
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
V23
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V32
()
FinSequence-like
FinSubsequence-like
rectangular
)
FinSequence
of ( ( ) ( non
empty
)
set
) ) holds
(
s
: ( (
rectangular
) ( non
empty
non
trivial
V19
()
V22
(
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
V23
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V32
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
rectangular
)
FinSequence
of ( ( ) ( non
empty
)
set
) )
/.
1 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
=
N-min
(
L~
s
: ( (
rectangular
) ( non
empty
non
trivial
V19
()
V22
(
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
V23
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V32
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
rectangular
)
FinSequence
of ( ( ) ( non
empty
)
set
) )
)
: ( ( ) (
compact
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) &
s
: ( (
rectangular
) ( non
empty
non
trivial
V19
()
V22
(
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
V23
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V32
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
rectangular
)
FinSequence
of ( ( ) ( non
empty
)
set
) )
/.
1 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
=
W-max
(
L~
s
: ( (
rectangular
) ( non
empty
non
trivial
V19
()
V22
(
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
V23
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V32
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
rectangular
)
FinSequence
of ( ( ) ( non
empty
)
set
) )
)
: ( ( ) (
compact
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) ) ;
theorem
:: SPRECT_1:84
for
s
being ( (
rectangular
) (
V19
()
V22
(
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
V23
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V32
()
FinSequence-like
FinSubsequence-like
rectangular
)
FinSequence
of ( ( ) ( non
empty
)
set
) ) holds
(
s
: ( (
rectangular
) ( non
empty
non
trivial
V19
()
V22
(
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
V23
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V32
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
rectangular
)
FinSequence
of ( ( ) ( non
empty
)
set
) )
/.
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
=
N-max
(
L~
s
: ( (
rectangular
) ( non
empty
non
trivial
V19
()
V22
(
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
V23
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V32
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
rectangular
)
FinSequence
of ( ( ) ( non
empty
)
set
) )
)
: ( ( ) (
compact
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) &
s
: ( (
rectangular
) ( non
empty
non
trivial
V19
()
V22
(
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
V23
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V32
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
rectangular
)
FinSequence
of ( ( ) ( non
empty
)
set
) )
/.
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
=
E-max
(
L~
s
: ( (
rectangular
) ( non
empty
non
trivial
V19
()
V22
(
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
V23
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V32
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
rectangular
)
FinSequence
of ( ( ) ( non
empty
)
set
) )
)
: ( ( ) (
compact
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) ) ;
theorem
:: SPRECT_1:85
for
s
being ( (
rectangular
) (
V19
()
V22
(
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
V23
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V32
()
FinSequence-like
FinSubsequence-like
rectangular
)
FinSequence
of ( ( ) ( non
empty
)
set
) ) holds
(
s
: ( (
rectangular
) ( non
empty
non
trivial
V19
()
V22
(
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
V23
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V32
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
rectangular
)
FinSequence
of ( ( ) ( non
empty
)
set
) )
/.
3 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
=
S-max
(
L~
s
: ( (
rectangular
) ( non
empty
non
trivial
V19
()
V22
(
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
V23
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V32
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
rectangular
)
FinSequence
of ( ( ) ( non
empty
)
set
) )
)
: ( ( ) (
compact
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) &
s
: ( (
rectangular
) ( non
empty
non
trivial
V19
()
V22
(
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
V23
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V32
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
rectangular
)
FinSequence
of ( ( ) ( non
empty
)
set
) )
/.
3 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
=
E-min
(
L~
s
: ( (
rectangular
) ( non
empty
non
trivial
V19
()
V22
(
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
V23
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V32
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
rectangular
)
FinSequence
of ( ( ) ( non
empty
)
set
) )
)
: ( ( ) (
compact
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) ) ;
theorem
:: SPRECT_1:86
for
s
being ( (
rectangular
) (
V19
()
V22
(
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
V23
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
V32
()
FinSequence-like
FinSubsequence-like
rectangular
)
FinSequence
of ( ( ) ( non
empty
)
set
) ) holds
(
s
: ( (
rectangular
) ( non
empty
non
trivial
V19
()
V22
(
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
V23
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V32
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
rectangular
)
FinSequence
of ( ( ) ( non
empty
)
set
) )
/.
4 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
=
S-min
(
L~
s
: ( (
rectangular
) ( non
empty
non
trivial
V19
()
V22
(
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
V23
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V32
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
rectangular
)
FinSequence
of ( ( ) ( non
empty
)
set
) )
)
: ( ( ) (
compact
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) &
s
: ( (
rectangular
) ( non
empty
non
trivial
V19
()
V22
(
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
V23
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V32
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
rectangular
)
FinSequence
of ( ( ) ( non
empty
)
set
) )
/.
4 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
=
W-min
(
L~
s
: ( (
rectangular
) ( non
empty
non
trivial
V19
()
V22
(
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
V23
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V32
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
rectangular
)
FinSequence
of ( ( ) ( non
empty
)
set
) )
)
: ( ( ) (
compact
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( 2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
-element
FinSequence-like
V191
() )
Element
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) ) ;
begin
theorem
:: SPRECT_1:87
for
r1
,
r2
,
s1
,
s2
being ( ( ) (
V11
()
real
ext-real
)
Real
) st
r1
: ( ( ) (
V11
()
real
ext-real
)
Real
)
<
r2
: ( ( ) (
V11
()
real
ext-real
)
Real
) &
s1
: ( ( ) (
V11
()
real
ext-real
)
Real
)
<
s2
: ( ( ) (
V11
()
real
ext-real
)
Real
) holds
[.
r1
: ( ( ) (
V11
()
real
ext-real
)
Real
) ,
r2
: ( ( ) (
V11
()
real
ext-real
)
Real
) ,
s1
: ( ( ) (
V11
()
real
ext-real
)
Real
) ,
s2
: ( ( ) (
V11
()
real
ext-real
)
Real
)
.]
: ( ( ) ( non
empty
compact
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) ) is
Jordan
;
registration
let
f
be ( (
rectangular
) ( non
empty
non
trivial
V19
()
V22
(
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
V23
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V32
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
rectangular
)
FinSequence
of ( ( ) ( non
empty
)
set
) ) ;
cluster
L~
f
: ( (
rectangular
) ( non
empty
non
trivial
V19
()
V22
(
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
V23
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V32
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
rectangular
)
FinSequence
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( non
empty
compact
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) )
->
Jordan
;
end;
definition
let
S
be ( ( ) ( )
Subset
of ) ;
redefine
attr
S
is
Jordan
means
:: SPRECT_1:def 3
(
S
: ( ( ) ( )
set
)
`
: ( ( ) ( )
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) )
<>
{}
: ( ( ) (
empty
trivial
Function-like
functional
FinSequence-membered
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
V245
() )
set
) & ex
A1
,
A2
being ( ( ) ( )
Subset
of ) st
(
S
: ( ( ) ( )
set
)
`
: ( ( ) ( )
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) )
=
A1
: ( ( ) ( )
Subset
of )
\/
A2
: ( ( ) ( )
Subset
of ) : ( ( ) ( )
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) ) &
A1
: ( ( ) ( )
Subset
of )
misses
A2
: ( ( ) ( )
Subset
of ) &
(
Cl
A1
: ( ( ) ( )
Subset
of )
)
: ( ( ) ( )
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) )
\
A1
: ( ( ) ( )
Subset
of ) : ( ( ) ( )
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) )
=
(
Cl
A2
: ( ( ) ( )
Subset
of )
)
: ( ( ) ( )
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) )
\
A2
: ( ( ) ( )
Subset
of ) : ( ( ) ( )
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) ) &
A1
: ( ( ) ( )
Subset
of )
is_a_component_of
S
: ( ( ) ( )
set
)
`
: ( ( ) ( )
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) ) &
A2
: ( ( ) ( )
Subset
of )
is_a_component_of
S
: ( ( ) ( )
set
)
`
: ( ( ) ( )
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) ) ) );
end;
theorem
:: SPRECT_1:88
for
f
being ( (
rectangular
) ( non
empty
non
trivial
V19
()
V22
(
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
V23
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V32
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
rectangular
)
FinSequence
of ( ( ) ( non
empty
)
set
) ) holds
LeftComp
f
: ( (
rectangular
) ( non
empty
non
trivial
V19
()
V22
(
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
V23
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V32
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
rectangular
)
FinSequence
of ( ( ) ( non
empty
)
set
) ) : ( ( ) ( )
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) )
misses
RightComp
f
: ( (
rectangular
) ( non
empty
non
trivial
V19
()
V22
(
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
V23
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V32
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
rectangular
)
FinSequence
of ( ( ) ( non
empty
)
set
) ) : ( ( ) ( )
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) ) ;
registration
let
f
be ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
) ( non
empty
non
trivial
V19
()
V22
(
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
V23
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V32
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
)
special_circular_sequence
) ;
cluster
LeftComp
f
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
) ( non
empty
non
trivial
V19
()
V22
(
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
V23
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V32
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
)
FinSequence
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( )
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) )
->
non
empty
;
cluster
RightComp
f
: ( ( non
empty
non
constant
circular
special
unfolded
s.c.c.
standard
) ( non
empty
non
trivial
V19
()
V22
(
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
V23
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V32
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
)
FinSequence
of the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( )
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) )
->
non
empty
;
end;
theorem
:: SPRECT_1:89
for
f
being ( (
rectangular
) ( non
empty
non
trivial
V19
()
V22
(
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
V23
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V32
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
rectangular
)
FinSequence
of ( ( ) ( non
empty
)
set
) ) holds
LeftComp
f
: ( (
rectangular
) ( non
empty
non
trivial
V19
()
V22
(
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
V23
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V32
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
rectangular
)
FinSequence
of ( ( ) ( non
empty
)
set
) ) : ( ( ) ( non
empty
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) )
<>
RightComp
f
: ( (
rectangular
) ( non
empty
non
trivial
V19
()
V22
(
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
V23
( the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) )
Function-like
non
constant
V32
()
FinSequence-like
FinSubsequence-like
circular
special
unfolded
s.c.c.
standard
rectangular
)
FinSequence
of ( ( ) ( non
empty
)
set
) ) : ( ( ) ( non
empty
)
Element
of
bool
the
carrier
of
(
TOP-REAL
2 : ( ( ) ( non
empty
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V97
()
V188
()
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
left_end
bounded_below
)
Element
of
NAT
: ( ( ) (
V199
()
V200
()
V201
()
V202
()
V203
()
V204
()
V205
()
bounded_below
)
Element
of
bool
REAL
: ( ( ) ( non
empty
V32
()
V199
()
V200
()
V201
()
V205
() non
bounded_below
non
bounded_above
V245
() )
set
) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
strict
) ( non
empty
TopSpace-like
V114
()
V145
()
V146
()
V147
()
V148
()
V149
()
V150
()
V151
()
strict
)
RLTopStruct
) : ( ( ) ( non
empty
)
set
) : ( ( ) ( non
empty
)
set
) ) ;