:: NFCONT_4 semantic presentation
begin
definition
let
n
be ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) ;
let
f
be ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,) ;
let
x0
be ( (
real
) (
V11
()
real
ext-real
)
number
) ;
pred
f
is_continuous_in
x0
means
:: NFCONT_4:def 1
ex
g
being ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
the
carrier
of
(
REAL-NS
n
: ( ( ) ( )
NORMSTR
)
)
: ( ( non
empty
strict
) ( non
empty
strict
)
NORMSTR
) : ( ( ) ( non
empty
)
set
)
-valued
Function-like
)
PartFunc
of ,) st
(
f
: ( ( ) ( )
Element
of
n
: ( ( ) ( )
NORMSTR
) )
=
g
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
the
carrier
of
(
REAL-NS
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( non
empty
strict
) ( non
empty
V106
()
V131
()
V132
()
V133
()
V134
()
V135
()
V136
()
V137
()
V141
()
V142
()
strict
RealNormSpace-like
V174
() )
NORMSTR
) : ( ( ) ( non
empty
)
set
)
-valued
Function-like
)
PartFunc
of ,) &
g
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
the
carrier
of
(
REAL-NS
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( non
empty
strict
) ( non
empty
V106
()
V131
()
V132
()
V133
()
V134
()
V135
()
V136
()
V137
()
V141
()
V142
()
strict
RealNormSpace-like
V174
() )
NORMSTR
) : ( ( ) ( non
empty
)
set
)
-valued
Function-like
)
PartFunc
of ,)
is_continuous_in
x0
: ( (
Function-like
quasi_total
) (
Relation-like
[:
n
: ( ( ) ( )
NORMSTR
) ,
n
: ( ( ) ( )
NORMSTR
)
:]
: ( ( ) ( )
set
)
-defined
n
: ( ( ) ( )
NORMSTR
)
-valued
Function-like
quasi_total
)
Element
of
bool
[:
[:
n
: ( ( ) ( )
NORMSTR
) ,
n
: ( ( ) ( )
NORMSTR
)
:]
: ( ( ) ( )
set
) ,
n
: ( ( ) ( )
NORMSTR
)
:]
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) );
end;
theorem
:: NFCONT_4:1
for
n
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
for
x0
being ( (
real
) (
V11
()
real
ext-real
)
number
)
for
f
being ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,)
for
h
being ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
the
carrier
of
(
REAL-NS
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( non
empty
strict
) ( non
empty
V106
()
V131
()
V132
()
V133
()
V134
()
V135
()
V136
()
V137
()
V141
()
V142
()
strict
RealNormSpace-like
V174
() )
NORMSTR
) : ( ( ) ( non
empty
)
set
)
-valued
Function-like
)
PartFunc
of ,) st
h
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
the
carrier
of
(
REAL-NS
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( non
empty
strict
) ( non
empty
V106
()
V131
()
V132
()
V133
()
V134
()
V135
()
V136
()
V137
()
V141
()
V142
()
strict
RealNormSpace-like
V174
() )
NORMSTR
) : ( ( ) ( non
empty
)
set
)
-valued
Function-like
)
PartFunc
of ,)
=
f
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,) holds
(
f
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,)
is_continuous_in
x0
: ( (
real
) (
V11
()
real
ext-real
)
number
) iff
h
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
the
carrier
of
(
REAL-NS
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( non
empty
strict
) ( non
empty
V106
()
V131
()
V132
()
V133
()
V134
()
V135
()
V136
()
V137
()
V141
()
V142
()
strict
RealNormSpace-like
V174
() )
NORMSTR
) : ( ( ) ( non
empty
)
set
)
-valued
Function-like
)
PartFunc
of ,)
is_continuous_in
x0
: ( (
real
) (
V11
()
real
ext-real
)
number
) ) ;
theorem
:: NFCONT_4:2
for
n
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
for
X
being ( ( ) ( )
set
)
for
x0
being ( (
real
) (
V11
()
real
ext-real
)
number
)
for
f
being ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,) st
x0
: ( (
real
) (
V11
()
real
ext-real
)
number
)
in
X
: ( ( ) ( )
set
) &
f
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,)
is_continuous_in
x0
: ( (
real
) (
V11
()
real
ext-real
)
number
) holds
f
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,)
|
X
: ( ( ) ( )
set
) : ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
Element
of
bool
[:
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ,
(
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
:]
: ( ( ) ( )
set
) : ( ( ) ( )
set
) )
is_continuous_in
x0
: ( (
real
) (
V11
()
real
ext-real
)
number
) ;
theorem
:: NFCONT_4:3
for
n
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
for
x0
being ( (
real
) (
V11
()
real
ext-real
)
number
)
for
f
being ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,) holds
(
f
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,)
is_continuous_in
x0
: ( (
real
) (
V11
()
real
ext-real
)
number
) iff (
x0
: ( (
real
) (
V11
()
real
ext-real
)
number
)
in
dom
f
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,) : ( ( ) ( )
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) & ( for
r
being ( ( ) (
V11
()
real
ext-real
)
Real
) st
0
: ( ( ) (
empty
trivial
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V11
()
real
ext-real
non
positive
non
negative
Function-like
functional
FinSequence-membered
V227
()
V228
()
V229
()
V230
()
V231
()
V232
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
<
r
: ( ( ) (
V11
()
real
ext-real
)
Real
) holds
ex
s
being ( (
real
) (
V11
()
real
ext-real
)
number
) st
(
0
: ( ( ) (
empty
trivial
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V11
()
real
ext-real
non
positive
non
negative
Function-like
functional
FinSequence-membered
V227
()
V228
()
V229
()
V230
()
V231
()
V232
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
<
s
: ( (
real
) (
V11
()
real
ext-real
)
number
) & ( for
x1
being ( (
real
) (
V11
()
real
ext-real
)
number
) st
x1
: ( (
real
) (
V11
()
real
ext-real
)
number
)
in
dom
f
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,) : ( ( ) ( )
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) &
abs
(
x1
: ( (
real
) (
V11
()
real
ext-real
)
number
)
-
x0
: ( (
real
) (
V11
()
real
ext-real
)
number
)
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) )
<
s
: ( (
real
) (
V11
()
real
ext-real
)
number
) holds
|.
(
(
f
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,)
/.
x1
: ( (
real
) (
V11
()
real
ext-real
)
number
)
)
: ( ( ) (
Relation-like
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) )
-defined
Function-like
V33
()
V34
()
V35
()
finite
V71
(
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
FinSubsequence-like
)
Element
of
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) )) )
-
(
f
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,)
/.
x0
: ( (
real
) (
V11
()
real
ext-real
)
number
)
)
: ( ( ) (
Relation-like
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) )
-defined
Function-like
V33
()
V34
()
V35
()
finite
V71
(
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
FinSubsequence-like
)
Element
of
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) )) )
)
: ( ( ) (
Relation-like
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) )
-defined
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-valued
Function-like
V33
()
V34
()
V35
()
finite
V71
(
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
FinSubsequence-like
)
M13
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ,
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) )) ))
.|
: ( ( ) (
V11
()
real
ext-real
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) )
<
r
: ( ( ) (
V11
()
real
ext-real
)
Real
) ) ) ) ) ) ;
theorem
:: NFCONT_4:4
for
n
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
for
r
being ( ( ) (
V11
()
real
ext-real
)
Real
)
for
z
being ( ( ) (
Relation-like
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) )
-defined
Function-like
V33
()
V34
()
V35
()
finite
V71
(
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
FinSubsequence-like
)
Element
of
REAL
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) )) )
for
w
being ( ( ) ( )
Point
of ( ( ) ( non
empty
)
set
) ) st
z
: ( ( ) (
Relation-like
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) )
-defined
Function-like
V33
()
V34
()
V35
()
finite
V71
(
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
FinSubsequence-like
)
Element
of
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) )) )
=
w
: ( ( ) ( )
Point
of ( ( ) ( non
empty
)
set
) ) holds
{
y
: ( ( ) ( )
Point
of ( ( ) ( non
empty
)
set
) ) where
y
is ( ( ) (
Relation-like
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) )
-defined
Function-like
V33
()
V34
()
V35
()
finite
V71
(
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
FinSubsequence-like
)
Element
of
REAL
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) )) ) :
|.
(
y
: ( ( ) ( )
Point
of ( ( ) ( non
empty
)
set
) )
-
z
: ( ( ) (
Relation-like
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) )
-defined
Function-like
V33
()
V34
()
V35
()
finite
V71
(
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
FinSubsequence-like
)
Element
of
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) )) )
)
: ( ( ) (
Relation-like
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) )
-defined
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-valued
Function-like
V33
()
V34
()
V35
()
finite
V71
(
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
FinSubsequence-like
)
M13
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ,
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) )) ))
.|
: ( ( ) (
V11
()
real
ext-real
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) )
<
r
: ( ( ) (
V11
()
real
ext-real
)
Real
)
}
=
{
y
: ( ( ) ( )
Point
of ( ( ) ( non
empty
)
set
) ) where
y
is ( ( ) ( )
Point
of ( ( ) ( non
empty
)
set
) ) :
||.
(
y
: ( ( ) ( )
Point
of ( ( ) ( non
empty
)
set
) )
-
w
: ( ( ) ( )
Point
of ( ( ) ( non
empty
)
set
) )
)
: ( ( ) ( )
Element
of the
carrier
of
(
REAL-NS
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( non
empty
strict
) ( non
empty
V106
()
V131
()
V132
()
V133
()
V134
()
V135
()
V136
()
V137
()
V141
()
V142
()
strict
RealNormSpace-like
V174
() )
NORMSTR
) : ( ( ) ( non
empty
)
set
) )
.||
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) )
<
r
: ( ( ) (
V11
()
real
ext-real
)
Real
)
}
;
definition
let
n
be ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) ;
let
Z
be ( ( ) ( )
set
) ;
let
f
be ( (
Function-like
) (
Relation-like
Z
: ( ( ) ( )
set
)
-defined
REAL
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,) ;
func
|.
f
.|
->
( (
Function-like
) (
Relation-like
Z
: ( ( ) ( )
Element
of
n
: ( ( ) ( )
NORMSTR
) )
-defined
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-valued
Function-like
V33
()
V34
()
V35
() )
PartFunc
of ,)
means
:: NFCONT_4:def 2
(
dom
it
: ( (
Function-like
quasi_total
) (
Relation-like
[:
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ,
n
: ( ( ) ( )
NORMSTR
)
:]
: ( ( ) ( )
set
)
-defined
n
: ( ( ) ( )
NORMSTR
)
-valued
Function-like
quasi_total
)
Element
of
bool
[:
[:
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ,
n
: ( ( ) ( )
NORMSTR
)
:]
: ( ( ) ( )
set
) ,
n
: ( ( ) ( )
NORMSTR
)
:]
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) : ( ( ) (
Relation-like
)
Element
of
bool
Z
: ( ( ) ( )
Element
of
n
: ( ( ) ( )
NORMSTR
) ) : ( ( ) ( )
set
) )
=
dom
f
: ( (
Function-like
quasi_total
) (
Relation-like
[:
n
: ( ( ) ( )
NORMSTR
) ,
n
: ( ( ) ( )
NORMSTR
)
:]
: ( ( ) ( )
set
)
-defined
n
: ( ( ) ( )
NORMSTR
)
-valued
Function-like
quasi_total
)
Element
of
bool
[:
[:
n
: ( ( ) ( )
NORMSTR
) ,
n
: ( ( ) ( )
NORMSTR
)
:]
: ( ( ) ( )
set
) ,
n
: ( ( ) ( )
NORMSTR
)
:]
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) : ( ( ) (
Relation-like
)
Element
of
bool
Z
: ( ( ) ( )
Element
of
n
: ( ( ) ( )
NORMSTR
) ) : ( ( ) ( )
set
) ) & ( for
x
being ( ( ) ( )
set
) st
x
: ( ( ) ( )
set
)
in
dom
it
: ( (
Function-like
quasi_total
) (
Relation-like
[:
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ,
n
: ( ( ) ( )
NORMSTR
)
:]
: ( ( ) ( )
set
)
-defined
n
: ( ( ) ( )
NORMSTR
)
-valued
Function-like
quasi_total
)
Element
of
bool
[:
[:
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ,
n
: ( ( ) ( )
NORMSTR
)
:]
: ( ( ) ( )
set
) ,
n
: ( ( ) ( )
NORMSTR
)
:]
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) : ( ( ) (
Relation-like
)
Element
of
bool
Z
: ( ( ) ( )
Element
of
n
: ( ( ) ( )
NORMSTR
) ) : ( ( ) ( )
set
) ) holds
it
: ( (
Function-like
quasi_total
) (
Relation-like
[:
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ,
n
: ( ( ) ( )
NORMSTR
)
:]
: ( ( ) ( )
set
)
-defined
n
: ( ( ) ( )
NORMSTR
)
-valued
Function-like
quasi_total
)
Element
of
bool
[:
[:
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ,
n
: ( ( ) ( )
NORMSTR
)
:]
: ( ( ) ( )
set
) ,
n
: ( ( ) ( )
NORMSTR
)
:]
: ( ( ) ( )
set
) : ( ( ) ( )
set
) )
/.
x
: ( ( ) ( )
set
) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) )
=
|.
(
f
: ( (
Function-like
quasi_total
) (
Relation-like
[:
n
: ( ( ) ( )
NORMSTR
) ,
n
: ( ( ) ( )
NORMSTR
)
:]
: ( ( ) ( )
set
)
-defined
n
: ( ( ) ( )
NORMSTR
)
-valued
Function-like
quasi_total
)
Element
of
bool
[:
[:
n
: ( ( ) ( )
NORMSTR
) ,
n
: ( ( ) ( )
NORMSTR
)
:]
: ( ( ) ( )
set
) ,
n
: ( ( ) ( )
NORMSTR
)
:]
: ( ( ) ( )
set
) : ( ( ) ( )
set
) )
/.
x
: ( ( ) ( )
set
)
)
: ( ( ) ( )
Element
of
REAL
n
: ( ( ) ( )
NORMSTR
) : ( ( ) ( )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) )) )
.|
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ) ) );
end;
definition
let
n
be ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) ;
let
Z
be ( ( non
empty
) ( non
empty
)
set
) ;
let
f
be ( (
Function-like
) (
Relation-like
Z
: ( ( non
empty
) ( non
empty
)
set
)
-defined
REAL
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,) ;
func
-
f
->
( (
Function-like
) (
Relation-like
Z
: ( ( ) ( )
Element
of
n
: ( ( ) ( )
NORMSTR
) )
-defined
REAL
n
: ( ( ) ( )
NORMSTR
) : ( ( ) ( )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,)
means
:: NFCONT_4:def 3
(
dom
it
: ( (
Function-like
quasi_total
) (
Relation-like
[:
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ,
n
: ( ( ) ( )
NORMSTR
)
:]
: ( ( ) ( )
set
)
-defined
n
: ( ( ) ( )
NORMSTR
)
-valued
Function-like
quasi_total
)
Element
of
bool
[:
[:
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ,
n
: ( ( ) ( )
NORMSTR
)
:]
: ( ( ) ( )
set
) ,
n
: ( ( ) ( )
NORMSTR
)
:]
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) : ( ( ) (
Relation-like
)
Element
of
bool
Z
: ( ( ) ( )
Element
of
n
: ( ( ) ( )
NORMSTR
) ) : ( ( ) ( )
set
) )
=
dom
f
: ( (
Function-like
quasi_total
) (
Relation-like
[:
n
: ( ( ) ( )
NORMSTR
) ,
n
: ( ( ) ( )
NORMSTR
)
:]
: ( ( ) ( )
set
)
-defined
n
: ( ( ) ( )
NORMSTR
)
-valued
Function-like
quasi_total
)
Element
of
bool
[:
[:
n
: ( ( ) ( )
NORMSTR
) ,
n
: ( ( ) ( )
NORMSTR
)
:]
: ( ( ) ( )
set
) ,
n
: ( ( ) ( )
NORMSTR
)
:]
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) : ( ( ) (
Relation-like
)
Element
of
bool
Z
: ( ( ) ( )
Element
of
n
: ( ( ) ( )
NORMSTR
) ) : ( ( ) ( )
set
) ) & ( for
c
being ( ( ) ( )
set
) st
c
: ( ( ) ( )
set
)
in
dom
it
: ( (
Function-like
quasi_total
) (
Relation-like
[:
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ,
n
: ( ( ) ( )
NORMSTR
)
:]
: ( ( ) ( )
set
)
-defined
n
: ( ( ) ( )
NORMSTR
)
-valued
Function-like
quasi_total
)
Element
of
bool
[:
[:
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ,
n
: ( ( ) ( )
NORMSTR
)
:]
: ( ( ) ( )
set
) ,
n
: ( ( ) ( )
NORMSTR
)
:]
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) : ( ( ) (
Relation-like
)
Element
of
bool
Z
: ( ( ) ( )
Element
of
n
: ( ( ) ( )
NORMSTR
) ) : ( ( ) ( )
set
) ) holds
it
: ( (
Function-like
quasi_total
) (
Relation-like
[:
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ,
n
: ( ( ) ( )
NORMSTR
)
:]
: ( ( ) ( )
set
)
-defined
n
: ( ( ) ( )
NORMSTR
)
-valued
Function-like
quasi_total
)
Element
of
bool
[:
[:
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ,
n
: ( ( ) ( )
NORMSTR
)
:]
: ( ( ) ( )
set
) ,
n
: ( ( ) ( )
NORMSTR
)
:]
: ( ( ) ( )
set
) : ( ( ) ( )
set
) )
/.
c
: ( ( ) ( )
set
) : ( ( ) ( )
Element
of
REAL
n
: ( ( ) ( )
NORMSTR
) : ( ( ) ( )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) )) )
=
-
(
f
: ( (
Function-like
quasi_total
) (
Relation-like
[:
n
: ( ( ) ( )
NORMSTR
) ,
n
: ( ( ) ( )
NORMSTR
)
:]
: ( ( ) ( )
set
)
-defined
n
: ( ( ) ( )
NORMSTR
)
-valued
Function-like
quasi_total
)
Element
of
bool
[:
[:
n
: ( ( ) ( )
NORMSTR
) ,
n
: ( ( ) ( )
NORMSTR
)
:]
: ( ( ) ( )
set
) ,
n
: ( ( ) ( )
NORMSTR
)
:]
: ( ( ) ( )
set
) : ( ( ) ( )
set
) )
/.
c
: ( ( ) ( )
set
)
)
: ( ( ) ( )
Element
of
REAL
n
: ( ( ) ( )
NORMSTR
) : ( ( ) ( )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) )) ) : ( ( ) (
Relation-like
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) )
-defined
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-valued
Function-like
V33
()
V34
()
V35
()
finite
V71
(
n
: ( ( ) ( )
NORMSTR
) )
FinSequence-like
FinSubsequence-like
)
M13
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ,
REAL
n
: ( ( ) ( )
NORMSTR
) : ( ( ) ( )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) )) )) ) );
end;
theorem
:: NFCONT_4:5
for
n
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
for
W
being ( ( non
empty
) ( non
empty
)
set
)
for
f1
,
f2
being ( (
Function-like
) (
Relation-like
b
2
: ( ( non
empty
) ( non
empty
)
set
)
-defined
the
carrier
of
(
REAL-NS
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( non
empty
strict
) ( non
empty
V106
()
V131
()
V132
()
V133
()
V134
()
V135
()
V136
()
V137
()
V141
()
V142
()
strict
RealNormSpace-like
V174
() )
NORMSTR
) : ( ( ) ( non
empty
)
set
)
-valued
Function-like
)
PartFunc
of ,)
for
g1
,
g2
being ( (
Function-like
) (
Relation-like
b
2
: ( ( non
empty
) ( non
empty
)
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,) st
f1
: ( (
Function-like
) (
Relation-like
b
2
: ( ( non
empty
) ( non
empty
)
set
)
-defined
the
carrier
of
(
REAL-NS
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( non
empty
strict
) ( non
empty
V106
()
V131
()
V132
()
V133
()
V134
()
V135
()
V136
()
V137
()
V141
()
V142
()
strict
RealNormSpace-like
V174
() )
NORMSTR
) : ( ( ) ( non
empty
)
set
)
-valued
Function-like
)
PartFunc
of ,)
=
g1
: ( (
Function-like
) (
Relation-like
b
2
: ( ( non
empty
) ( non
empty
)
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,) &
f2
: ( (
Function-like
) (
Relation-like
b
2
: ( ( non
empty
) ( non
empty
)
set
)
-defined
the
carrier
of
(
REAL-NS
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( non
empty
strict
) ( non
empty
V106
()
V131
()
V132
()
V133
()
V134
()
V135
()
V136
()
V137
()
V141
()
V142
()
strict
RealNormSpace-like
V174
() )
NORMSTR
) : ( ( ) ( non
empty
)
set
)
-valued
Function-like
)
PartFunc
of ,)
=
g2
: ( (
Function-like
) (
Relation-like
b
2
: ( ( non
empty
) ( non
empty
)
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,) holds
f1
: ( (
Function-like
) (
Relation-like
b
2
: ( ( non
empty
) ( non
empty
)
set
)
-defined
the
carrier
of
(
REAL-NS
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( non
empty
strict
) ( non
empty
V106
()
V131
()
V132
()
V133
()
V134
()
V135
()
V136
()
V137
()
V141
()
V142
()
strict
RealNormSpace-like
V174
() )
NORMSTR
) : ( ( ) ( non
empty
)
set
)
-valued
Function-like
)
PartFunc
of ,)
+
f2
: ( (
Function-like
) (
Relation-like
b
2
: ( ( non
empty
) ( non
empty
)
set
)
-defined
the
carrier
of
(
REAL-NS
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( non
empty
strict
) ( non
empty
V106
()
V131
()
V132
()
V133
()
V134
()
V135
()
V136
()
V137
()
V141
()
V142
()
strict
RealNormSpace-like
V174
() )
NORMSTR
) : ( ( ) ( non
empty
)
set
)
-valued
Function-like
)
PartFunc
of ,) : ( (
Function-like
) (
Relation-like
b
2
: ( ( non
empty
) ( non
empty
)
set
)
-defined
the
carrier
of
(
REAL-NS
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( non
empty
strict
) ( non
empty
V106
()
V131
()
V132
()
V133
()
V134
()
V135
()
V136
()
V137
()
V141
()
V142
()
strict
RealNormSpace-like
V174
() )
NORMSTR
) : ( ( ) ( non
empty
)
set
)
-valued
Function-like
)
Element
of
bool
[:
b
2
: ( ( non
empty
) ( non
empty
)
set
) , the
carrier
of
(
REAL-NS
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( non
empty
strict
) ( non
empty
V106
()
V131
()
V132
()
V133
()
V134
()
V135
()
V136
()
V137
()
V141
()
V142
()
strict
RealNormSpace-like
V174
() )
NORMSTR
) : ( ( ) ( non
empty
)
set
)
:]
: ( ( ) ( )
set
) : ( ( ) ( )
set
) )
=
g1
: ( (
Function-like
) (
Relation-like
b
2
: ( ( non
empty
) ( non
empty
)
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,)
+
g2
: ( (
Function-like
) (
Relation-like
b
2
: ( ( non
empty
) ( non
empty
)
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,) : ( (
Function-like
) (
Relation-like
b
2
: ( ( non
empty
) ( non
empty
)
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
Element
of
bool
[:
b
2
: ( ( non
empty
) ( non
empty
)
set
) ,
(
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
:]
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ;
theorem
:: NFCONT_4:6
for
n
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
for
W
being ( ( non
empty
) ( non
empty
)
set
)
for
f1
being ( (
Function-like
) (
Relation-like
b
2
: ( ( non
empty
) ( non
empty
)
set
)
-defined
the
carrier
of
(
REAL-NS
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( non
empty
strict
) ( non
empty
V106
()
V131
()
V132
()
V133
()
V134
()
V135
()
V136
()
V137
()
V141
()
V142
()
strict
RealNormSpace-like
V174
() )
NORMSTR
) : ( ( ) ( non
empty
)
set
)
-valued
Function-like
)
PartFunc
of ,)
for
g1
being ( (
Function-like
) (
Relation-like
b
2
: ( ( non
empty
) ( non
empty
)
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,)
for
a
being ( ( ) (
V11
()
real
ext-real
)
Real
) st
f1
: ( (
Function-like
) (
Relation-like
b
2
: ( ( non
empty
) ( non
empty
)
set
)
-defined
the
carrier
of
(
REAL-NS
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( non
empty
strict
) ( non
empty
V106
()
V131
()
V132
()
V133
()
V134
()
V135
()
V136
()
V137
()
V141
()
V142
()
strict
RealNormSpace-like
V174
() )
NORMSTR
) : ( ( ) ( non
empty
)
set
)
-valued
Function-like
)
PartFunc
of ,)
=
g1
: ( (
Function-like
) (
Relation-like
b
2
: ( ( non
empty
) ( non
empty
)
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,) holds
a
: ( ( ) (
V11
()
real
ext-real
)
Real
)
(#)
f1
: ( (
Function-like
) (
Relation-like
b
2
: ( ( non
empty
) ( non
empty
)
set
)
-defined
the
carrier
of
(
REAL-NS
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( non
empty
strict
) ( non
empty
V106
()
V131
()
V132
()
V133
()
V134
()
V135
()
V136
()
V137
()
V141
()
V142
()
strict
RealNormSpace-like
V174
() )
NORMSTR
) : ( ( ) ( non
empty
)
set
)
-valued
Function-like
)
PartFunc
of ,) : ( (
Function-like
) (
Relation-like
b
2
: ( ( non
empty
) ( non
empty
)
set
)
-defined
the
carrier
of
(
REAL-NS
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( non
empty
strict
) ( non
empty
V106
()
V131
()
V132
()
V133
()
V134
()
V135
()
V136
()
V137
()
V141
()
V142
()
strict
RealNormSpace-like
V174
() )
NORMSTR
) : ( ( ) ( non
empty
)
set
)
-valued
Function-like
)
Element
of
bool
[:
b
2
: ( ( non
empty
) ( non
empty
)
set
) , the
carrier
of
(
REAL-NS
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( non
empty
strict
) ( non
empty
V106
()
V131
()
V132
()
V133
()
V134
()
V135
()
V136
()
V137
()
V141
()
V142
()
strict
RealNormSpace-like
V174
() )
NORMSTR
) : ( ( ) ( non
empty
)
set
)
:]
: ( ( ) ( )
set
) : ( ( ) ( )
set
) )
=
a
: ( ( ) (
V11
()
real
ext-real
)
Real
)
(#)
g1
: ( (
Function-like
) (
Relation-like
b
2
: ( ( non
empty
) ( non
empty
)
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,) : ( (
Function-like
) (
Relation-like
b
2
: ( ( non
empty
) ( non
empty
)
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
Element
of
bool
[:
b
2
: ( ( non
empty
) ( non
empty
)
set
) ,
(
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
:]
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ;
theorem
:: NFCONT_4:7
for
n
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
for
W
being ( ( non
empty
) ( non
empty
)
set
)
for
f1
being ( (
Function-like
) (
Relation-like
b
2
: ( ( non
empty
) ( non
empty
)
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,) holds
(
-
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) (
V11
()
real
ext-real
non
positive
)
Element
of
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) )
(#)
f1
: ( (
Function-like
) (
Relation-like
b
2
: ( ( non
empty
) ( non
empty
)
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,) : ( (
Function-like
) (
Relation-like
b
2
: ( ( non
empty
) ( non
empty
)
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
Element
of
bool
[:
b
2
: ( ( non
empty
) ( non
empty
)
set
) ,
(
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
:]
: ( ( ) ( )
set
) : ( ( ) ( )
set
) )
=
-
f1
: ( (
Function-like
) (
Relation-like
b
2
: ( ( non
empty
) ( non
empty
)
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,) : ( (
Function-like
) (
Relation-like
b
2
: ( ( non
empty
) ( non
empty
)
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,) ;
theorem
:: NFCONT_4:8
for
n
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
for
W
being ( ( non
empty
) ( non
empty
)
set
)
for
f1
being ( (
Function-like
) (
Relation-like
b
2
: ( ( non
empty
) ( non
empty
)
set
)
-defined
the
carrier
of
(
REAL-NS
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( non
empty
strict
) ( non
empty
V106
()
V131
()
V132
()
V133
()
V134
()
V135
()
V136
()
V137
()
V141
()
V142
()
strict
RealNormSpace-like
V174
() )
NORMSTR
) : ( ( ) ( non
empty
)
set
)
-valued
Function-like
)
PartFunc
of ,)
for
g1
being ( (
Function-like
) (
Relation-like
b
2
: ( ( non
empty
) ( non
empty
)
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,) st
f1
: ( (
Function-like
) (
Relation-like
b
2
: ( ( non
empty
) ( non
empty
)
set
)
-defined
the
carrier
of
(
REAL-NS
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( non
empty
strict
) ( non
empty
V106
()
V131
()
V132
()
V133
()
V134
()
V135
()
V136
()
V137
()
V141
()
V142
()
strict
RealNormSpace-like
V174
() )
NORMSTR
) : ( ( ) ( non
empty
)
set
)
-valued
Function-like
)
PartFunc
of ,)
=
g1
: ( (
Function-like
) (
Relation-like
b
2
: ( ( non
empty
) ( non
empty
)
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,) holds
-
f1
: ( (
Function-like
) (
Relation-like
b
2
: ( ( non
empty
) ( non
empty
)
set
)
-defined
the
carrier
of
(
REAL-NS
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( non
empty
strict
) ( non
empty
V106
()
V131
()
V132
()
V133
()
V134
()
V135
()
V136
()
V137
()
V141
()
V142
()
strict
RealNormSpace-like
V174
() )
NORMSTR
) : ( ( ) ( non
empty
)
set
)
-valued
Function-like
)
PartFunc
of ,) : ( (
Function-like
) (
Relation-like
b
2
: ( ( non
empty
) ( non
empty
)
set
)
-defined
the
carrier
of
(
REAL-NS
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( non
empty
strict
) ( non
empty
V106
()
V131
()
V132
()
V133
()
V134
()
V135
()
V136
()
V137
()
V141
()
V142
()
strict
RealNormSpace-like
V174
() )
NORMSTR
) : ( ( ) ( non
empty
)
set
)
-valued
Function-like
)
Element
of
bool
[:
b
2
: ( ( non
empty
) ( non
empty
)
set
) , the
carrier
of
(
REAL-NS
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( non
empty
strict
) ( non
empty
V106
()
V131
()
V132
()
V133
()
V134
()
V135
()
V136
()
V137
()
V141
()
V142
()
strict
RealNormSpace-like
V174
() )
NORMSTR
) : ( ( ) ( non
empty
)
set
)
:]
: ( ( ) ( )
set
) : ( ( ) ( )
set
) )
=
-
g1
: ( (
Function-like
) (
Relation-like
b
2
: ( ( non
empty
) ( non
empty
)
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,) : ( (
Function-like
) (
Relation-like
b
2
: ( ( non
empty
) ( non
empty
)
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,) ;
theorem
:: NFCONT_4:9
for
n
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
for
W
being ( ( non
empty
) ( non
empty
)
set
)
for
f1
being ( (
Function-like
) (
Relation-like
b
2
: ( ( non
empty
) ( non
empty
)
set
)
-defined
the
carrier
of
(
REAL-NS
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( non
empty
strict
) ( non
empty
V106
()
V131
()
V132
()
V133
()
V134
()
V135
()
V136
()
V137
()
V141
()
V142
()
strict
RealNormSpace-like
V174
() )
NORMSTR
) : ( ( ) ( non
empty
)
set
)
-valued
Function-like
)
PartFunc
of ,)
for
g1
being ( (
Function-like
) (
Relation-like
b
2
: ( ( non
empty
) ( non
empty
)
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,) st
f1
: ( (
Function-like
) (
Relation-like
b
2
: ( ( non
empty
) ( non
empty
)
set
)
-defined
the
carrier
of
(
REAL-NS
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( non
empty
strict
) ( non
empty
V106
()
V131
()
V132
()
V133
()
V134
()
V135
()
V136
()
V137
()
V141
()
V142
()
strict
RealNormSpace-like
V174
() )
NORMSTR
) : ( ( ) ( non
empty
)
set
)
-valued
Function-like
)
PartFunc
of ,)
=
g1
: ( (
Function-like
) (
Relation-like
b
2
: ( ( non
empty
) ( non
empty
)
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,) holds
||.
f1
: ( (
Function-like
) (
Relation-like
b
2
: ( ( non
empty
) ( non
empty
)
set
)
-defined
the
carrier
of
(
REAL-NS
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( non
empty
strict
) ( non
empty
V106
()
V131
()
V132
()
V133
()
V134
()
V135
()
V136
()
V137
()
V141
()
V142
()
strict
RealNormSpace-like
V174
() )
NORMSTR
) : ( ( ) ( non
empty
)
set
)
-valued
Function-like
)
PartFunc
of ,)
.||
: ( (
Function-like
) (
Relation-like
b
2
: ( ( non
empty
) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-valued
Function-like
V33
()
V34
()
V35
() )
Element
of
bool
[:
b
2
: ( ( non
empty
) ( non
empty
)
set
) ,
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
:]
: ( ( ) ( )
set
) : ( ( ) ( )
set
) )
=
|.
g1
: ( (
Function-like
) (
Relation-like
b
2
: ( ( non
empty
) ( non
empty
)
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,)
.|
: ( (
Function-like
) (
Relation-like
b
2
: ( ( non
empty
) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-valued
Function-like
V33
()
V34
()
V35
() )
PartFunc
of ,) ;
theorem
:: NFCONT_4:10
for
n
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
for
W
being ( ( non
empty
) ( non
empty
)
set
)
for
f1
,
f2
being ( (
Function-like
) (
Relation-like
b
2
: ( ( non
empty
) ( non
empty
)
set
)
-defined
the
carrier
of
(
REAL-NS
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( non
empty
strict
) ( non
empty
V106
()
V131
()
V132
()
V133
()
V134
()
V135
()
V136
()
V137
()
V141
()
V142
()
strict
RealNormSpace-like
V174
() )
NORMSTR
) : ( ( ) ( non
empty
)
set
)
-valued
Function-like
)
PartFunc
of ,)
for
g1
,
g2
being ( (
Function-like
) (
Relation-like
b
2
: ( ( non
empty
) ( non
empty
)
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,) st
f1
: ( (
Function-like
) (
Relation-like
b
2
: ( ( non
empty
) ( non
empty
)
set
)
-defined
the
carrier
of
(
REAL-NS
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( non
empty
strict
) ( non
empty
V106
()
V131
()
V132
()
V133
()
V134
()
V135
()
V136
()
V137
()
V141
()
V142
()
strict
RealNormSpace-like
V174
() )
NORMSTR
) : ( ( ) ( non
empty
)
set
)
-valued
Function-like
)
PartFunc
of ,)
=
g1
: ( (
Function-like
) (
Relation-like
b
2
: ( ( non
empty
) ( non
empty
)
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,) &
f2
: ( (
Function-like
) (
Relation-like
b
2
: ( ( non
empty
) ( non
empty
)
set
)
-defined
the
carrier
of
(
REAL-NS
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( non
empty
strict
) ( non
empty
V106
()
V131
()
V132
()
V133
()
V134
()
V135
()
V136
()
V137
()
V141
()
V142
()
strict
RealNormSpace-like
V174
() )
NORMSTR
) : ( ( ) ( non
empty
)
set
)
-valued
Function-like
)
PartFunc
of ,)
=
g2
: ( (
Function-like
) (
Relation-like
b
2
: ( ( non
empty
) ( non
empty
)
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,) holds
f1
: ( (
Function-like
) (
Relation-like
b
2
: ( ( non
empty
) ( non
empty
)
set
)
-defined
the
carrier
of
(
REAL-NS
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( non
empty
strict
) ( non
empty
V106
()
V131
()
V132
()
V133
()
V134
()
V135
()
V136
()
V137
()
V141
()
V142
()
strict
RealNormSpace-like
V174
() )
NORMSTR
) : ( ( ) ( non
empty
)
set
)
-valued
Function-like
)
PartFunc
of ,)
-
f2
: ( (
Function-like
) (
Relation-like
b
2
: ( ( non
empty
) ( non
empty
)
set
)
-defined
the
carrier
of
(
REAL-NS
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( non
empty
strict
) ( non
empty
V106
()
V131
()
V132
()
V133
()
V134
()
V135
()
V136
()
V137
()
V141
()
V142
()
strict
RealNormSpace-like
V174
() )
NORMSTR
) : ( ( ) ( non
empty
)
set
)
-valued
Function-like
)
PartFunc
of ,) : ( (
Function-like
) (
Relation-like
b
2
: ( ( non
empty
) ( non
empty
)
set
)
-defined
the
carrier
of
(
REAL-NS
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( non
empty
strict
) ( non
empty
V106
()
V131
()
V132
()
V133
()
V134
()
V135
()
V136
()
V137
()
V141
()
V142
()
strict
RealNormSpace-like
V174
() )
NORMSTR
) : ( ( ) ( non
empty
)
set
)
-valued
Function-like
)
Element
of
bool
[:
b
2
: ( ( non
empty
) ( non
empty
)
set
) , the
carrier
of
(
REAL-NS
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( non
empty
strict
) ( non
empty
V106
()
V131
()
V132
()
V133
()
V134
()
V135
()
V136
()
V137
()
V141
()
V142
()
strict
RealNormSpace-like
V174
() )
NORMSTR
) : ( ( ) ( non
empty
)
set
)
:]
: ( ( ) ( )
set
) : ( ( ) ( )
set
) )
=
g1
: ( (
Function-like
) (
Relation-like
b
2
: ( ( non
empty
) ( non
empty
)
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,)
-
g2
: ( (
Function-like
) (
Relation-like
b
2
: ( ( non
empty
) ( non
empty
)
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,) : ( (
Function-like
) (
Relation-like
b
2
: ( ( non
empty
) ( non
empty
)
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
Element
of
bool
[:
b
2
: ( ( non
empty
) ( non
empty
)
set
) ,
(
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
:]
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ;
theorem
:: NFCONT_4:11
for
n
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
for
x0
being ( (
real
) (
V11
()
real
ext-real
)
number
)
for
f
being ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,) holds
(
f
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,)
is_continuous_in
x0
: ( (
real
) (
V11
()
real
ext-real
)
number
) iff (
x0
: ( (
real
) (
V11
()
real
ext-real
)
number
)
in
dom
f
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,) : ( ( ) ( )
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) & ( for
N1
being ( ( ) (
functional
FinSequence-membered
V227
()
V228
()
V229
() )
Subset
of ( ( ) ( )
set
) ) st ex
r
being ( ( ) (
V11
()
real
ext-real
)
Real
) st
(
0
: ( ( ) (
empty
trivial
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V11
()
real
ext-real
non
positive
non
negative
Function-like
functional
FinSequence-membered
V227
()
V228
()
V229
()
V230
()
V231
()
V232
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
<
r
: ( ( ) ( )
Neighbourhood
of
b
2
: ( (
real
) (
V11
()
real
ext-real
)
number
) ) &
{
y
: ( (
real
) (
V11
()
real
ext-real
)
number
) where
y
is ( ( ) (
Relation-like
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) )
-defined
Function-like
V33
()
V34
()
V35
()
finite
V71
(
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
FinSubsequence-like
)
Element
of
REAL
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) )) ) :
|.
(
y
: ( (
real
) (
V11
()
real
ext-real
)
number
)
-
(
f
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,)
/.
x0
: ( (
real
) (
V11
()
real
ext-real
)
number
)
)
: ( ( ) (
Relation-like
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) )
-defined
Function-like
V33
()
V34
()
V35
()
finite
V71
(
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
FinSubsequence-like
)
Element
of
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) )) )
)
: ( ( ) (
Relation-like
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) )
-defined
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-valued
Function-like
V33
()
V34
()
V35
()
finite
V71
(
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
FinSubsequence-like
)
M13
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ,
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) )) ))
.|
: ( ( ) (
V11
()
real
ext-real
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) )
<
r
: ( ( ) ( )
Neighbourhood
of
b
2
: ( (
real
) (
V11
()
real
ext-real
)
number
) )
}
=
N1
: ( ( ) (
functional
FinSequence-membered
V227
()
V228
()
V229
() )
Subset
of ( ( ) ( )
set
) ) ) holds
ex
N
being ( ( ) ( )
Neighbourhood
of
x0
: ( (
real
) (
V11
()
real
ext-real
)
number
) ) st
for
x1
being ( (
real
) (
V11
()
real
ext-real
)
number
) st
x1
: ( (
real
) (
V11
()
real
ext-real
)
number
)
in
dom
f
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,) : ( ( ) ( )
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) &
x1
: ( (
real
) (
V11
()
real
ext-real
)
number
)
in
N
: ( ( ) ( )
Neighbourhood
of
b
2
: ( (
real
) (
V11
()
real
ext-real
)
number
) ) holds
f
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,)
/.
x1
: ( (
real
) (
V11
()
real
ext-real
)
number
) : ( ( ) (
Relation-like
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) )
-defined
Function-like
V33
()
V34
()
V35
()
finite
V71
(
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
FinSubsequence-like
)
Element
of
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) )) )
in
N1
: ( ( ) (
functional
FinSequence-membered
V227
()
V228
()
V229
() )
Subset
of ( ( ) ( )
set
) ) ) ) ) ;
theorem
:: NFCONT_4:12
for
n
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
for
x0
being ( (
real
) (
V11
()
real
ext-real
)
number
)
for
f
being ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,) holds
(
f
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,)
is_continuous_in
x0
: ( (
real
) (
V11
()
real
ext-real
)
number
) iff (
x0
: ( (
real
) (
V11
()
real
ext-real
)
number
)
in
dom
f
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,) : ( ( ) ( )
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) & ( for
N1
being ( ( ) (
functional
FinSequence-membered
V227
()
V228
()
V229
() )
Subset
of ( ( ) ( )
set
) ) st ex
r
being ( ( ) (
V11
()
real
ext-real
)
Real
) st
(
0
: ( ( ) (
empty
trivial
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V11
()
real
ext-real
non
positive
non
negative
Function-like
functional
FinSequence-membered
V227
()
V228
()
V229
()
V230
()
V231
()
V232
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
<
r
: ( ( ) ( )
Neighbourhood
of
b
2
: ( (
real
) (
V11
()
real
ext-real
)
number
) ) &
{
y
: ( ( ) (
Relation-like
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) )
-defined
Function-like
V33
()
V34
()
V35
()
finite
V71
(
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
FinSubsequence-like
)
Element
of
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) )) ) where
y
is ( ( ) (
Relation-like
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) )
-defined
Function-like
V33
()
V34
()
V35
()
finite
V71
(
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
FinSubsequence-like
)
Element
of
REAL
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) )) ) :
|.
(
y
: ( ( ) (
Relation-like
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) )
-defined
Function-like
V33
()
V34
()
V35
()
finite
V71
(
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
FinSubsequence-like
)
Element
of
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) )) )
-
(
f
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,)
/.
x0
: ( (
real
) (
V11
()
real
ext-real
)
number
)
)
: ( ( ) (
Relation-like
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) )
-defined
Function-like
V33
()
V34
()
V35
()
finite
V71
(
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
FinSubsequence-like
)
Element
of
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) )) )
)
: ( ( ) (
Relation-like
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) )
-defined
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-valued
Function-like
V33
()
V34
()
V35
()
finite
V71
(
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
FinSubsequence-like
)
M13
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ,
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) )) ))
.|
: ( ( ) (
V11
()
real
ext-real
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) )
<
r
: ( ( ) ( )
Neighbourhood
of
b
2
: ( (
real
) (
V11
()
real
ext-real
)
number
) )
}
=
N1
: ( ( ) (
functional
FinSequence-membered
V227
()
V228
()
V229
() )
Subset
of ( ( ) ( )
set
) ) ) holds
ex
N
being ( ( ) ( )
Neighbourhood
of
x0
: ( (
real
) (
V11
()
real
ext-real
)
number
) ) st
f
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,)
.:
N
: ( ( ) ( )
Neighbourhood
of
b
2
: ( (
real
) (
V11
()
real
ext-real
)
number
) ) : ( ( ) (
functional
FinSequence-membered
V227
()
V228
()
V229
() )
Element
of
bool
(
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) )) : ( ( ) ( )
set
) )
c=
N1
: ( ( ) (
functional
FinSequence-membered
V227
()
V228
()
V229
() )
Subset
of ( ( ) ( )
set
) ) ) ) ) ;
theorem
:: NFCONT_4:13
for
n
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
for
x0
being ( (
real
) (
V11
()
real
ext-real
)
number
)
for
f
being ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,) st ex
N
being ( ( ) ( )
Neighbourhood
of
x0
: ( (
real
) (
V11
()
real
ext-real
)
number
) ) st
(
dom
f
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,)
)
: ( ( ) ( )
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) )
/\
N
: ( ( ) ( )
Neighbourhood
of
b
2
: ( (
real
) (
V11
()
real
ext-real
)
number
) ) : ( ( ) ( )
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) )
=
{
x0
: ( (
real
) (
V11
()
real
ext-real
)
number
)
}
: ( ( ) ( non
empty
trivial
)
set
) holds
f
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,)
is_continuous_in
x0
: ( (
real
) (
V11
()
real
ext-real
)
number
) ;
theorem
:: NFCONT_4:14
for
n
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
for
x0
being ( (
real
) (
V11
()
real
ext-real
)
number
)
for
f1
,
f2
being ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,) st
x0
: ( (
real
) (
V11
()
real
ext-real
)
number
)
in
(
dom
f1
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,)
)
: ( ( ) ( )
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) )
/\
(
dom
f2
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,)
)
: ( ( ) ( )
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) : ( ( ) ( )
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) &
f1
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,)
is_continuous_in
x0
: ( (
real
) (
V11
()
real
ext-real
)
number
) &
f2
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,)
is_continuous_in
x0
: ( (
real
) (
V11
()
real
ext-real
)
number
) holds
f1
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,)
+
f2
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,) : ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
Element
of
bool
[:
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ,
(
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
:]
: ( ( ) ( )
set
) : ( ( ) ( )
set
) )
is_continuous_in
x0
: ( (
real
) (
V11
()
real
ext-real
)
number
) ;
theorem
:: NFCONT_4:15
for
n
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
for
x0
being ( (
real
) (
V11
()
real
ext-real
)
number
)
for
f1
,
f2
being ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,) st
x0
: ( (
real
) (
V11
()
real
ext-real
)
number
)
in
(
dom
f1
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,)
)
: ( ( ) ( )
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) )
/\
(
dom
f2
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,)
)
: ( ( ) ( )
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) : ( ( ) ( )
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) &
f1
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,)
is_continuous_in
x0
: ( (
real
) (
V11
()
real
ext-real
)
number
) &
f2
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,)
is_continuous_in
x0
: ( (
real
) (
V11
()
real
ext-real
)
number
) holds
f1
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,)
-
f2
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,) : ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
Element
of
bool
[:
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ,
(
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
:]
: ( ( ) ( )
set
) : ( ( ) ( )
set
) )
is_continuous_in
x0
: ( (
real
) (
V11
()
real
ext-real
)
number
) ;
theorem
:: NFCONT_4:16
for
n
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
for
r
being ( ( ) (
V11
()
real
ext-real
)
Real
)
for
x0
being ( (
real
) (
V11
()
real
ext-real
)
number
)
for
f
being ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,) st
f
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,)
is_continuous_in
x0
: ( (
real
) (
V11
()
real
ext-real
)
number
) holds
r
: ( ( ) (
V11
()
real
ext-real
)
Real
)
(#)
f
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,) : ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
Element
of
bool
[:
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ,
(
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
:]
: ( ( ) ( )
set
) : ( ( ) ( )
set
) )
is_continuous_in
x0
: ( (
real
) (
V11
()
real
ext-real
)
number
) ;
theorem
:: NFCONT_4:17
for
n
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
for
x0
being ( (
real
) (
V11
()
real
ext-real
)
number
)
for
f
being ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,) st
x0
: ( (
real
) (
V11
()
real
ext-real
)
number
)
in
dom
f
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,) : ( ( ) ( )
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) &
f
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,)
is_continuous_in
x0
: ( (
real
) (
V11
()
real
ext-real
)
number
) holds
|.
f
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,)
.|
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-valued
Function-like
V33
()
V34
()
V35
() )
PartFunc
of ,)
is_continuous_in
x0
: ( (
real
) (
V11
()
real
ext-real
)
number
) ;
theorem
:: NFCONT_4:18
for
n
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
for
x0
being ( (
real
) (
V11
()
real
ext-real
)
number
)
for
f
being ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,) st
x0
: ( (
real
) (
V11
()
real
ext-real
)
number
)
in
dom
f
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,) : ( ( ) ( )
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) &
f
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,)
is_continuous_in
x0
: ( (
real
) (
V11
()
real
ext-real
)
number
) holds
-
f
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,) : ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,)
is_continuous_in
x0
: ( (
real
) (
V11
()
real
ext-real
)
number
) ;
theorem
:: NFCONT_4:19
for
n
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
for
x0
being ( (
real
) (
V11
()
real
ext-real
)
number
)
for
S
being ( ( non
empty
V106
()
V131
()
V132
()
V133
()
V134
()
V135
()
V136
()
V137
()
V141
()
V142
()
RealNormSpace-like
) ( non
empty
V106
()
V131
()
V132
()
V133
()
V134
()
V135
()
V136
()
V137
()
V141
()
V142
()
RealNormSpace-like
)
RealNormSpace
)
for
z
being ( ( ) ( )
Point
of ( ( ) ( non
empty
)
set
) )
for
f1
being ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,)
for
f2
being ( (
Function-like
) (
Relation-like
the
carrier
of
(
REAL-NS
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( non
empty
strict
) ( non
empty
V106
()
V131
()
V132
()
V133
()
V134
()
V135
()
V136
()
V137
()
V141
()
V142
()
strict
RealNormSpace-like
V174
() )
NORMSTR
) : ( ( ) ( non
empty
)
set
)
-defined
the
carrier
of
b
3
: ( ( non
empty
V106
()
V131
()
V132
()
V133
()
V134
()
V135
()
V136
()
V137
()
V141
()
V142
()
RealNormSpace-like
) ( non
empty
V106
()
V131
()
V132
()
V133
()
V134
()
V135
()
V136
()
V137
()
V141
()
V142
()
RealNormSpace-like
)
RealNormSpace
) : ( ( ) ( non
empty
)
set
)
-valued
Function-like
)
PartFunc
of ,) st
x0
: ( (
real
) (
V11
()
real
ext-real
)
number
)
in
dom
(
f2
: ( (
Function-like
) (
Relation-like
the
carrier
of
(
REAL-NS
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( non
empty
strict
) ( non
empty
V106
()
V131
()
V132
()
V133
()
V134
()
V135
()
V136
()
V137
()
V141
()
V142
()
strict
RealNormSpace-like
V174
() )
NORMSTR
) : ( ( ) ( non
empty
)
set
)
-defined
the
carrier
of
b
3
: ( ( non
empty
V106
()
V131
()
V132
()
V133
()
V134
()
V135
()
V136
()
V137
()
V141
()
V142
()
RealNormSpace-like
) ( non
empty
V106
()
V131
()
V132
()
V133
()
V134
()
V135
()
V136
()
V137
()
V141
()
V142
()
RealNormSpace-like
)
RealNormSpace
) : ( ( ) ( non
empty
)
set
)
-valued
Function-like
)
PartFunc
of ,)
*
f1
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,)
)
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
the
carrier
of
b
3
: ( ( non
empty
V106
()
V131
()
V132
()
V133
()
V134
()
V135
()
V136
()
V137
()
V141
()
V142
()
RealNormSpace-like
) ( non
empty
V106
()
V131
()
V132
()
V133
()
V134
()
V135
()
V136
()
V137
()
V141
()
V142
()
RealNormSpace-like
)
RealNormSpace
) : ( ( ) ( non
empty
)
set
)
-valued
Function-like
)
Element
of
bool
[:
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) , the
carrier
of
b
3
: ( ( non
empty
V106
()
V131
()
V132
()
V133
()
V134
()
V135
()
V136
()
V137
()
V141
()
V142
()
RealNormSpace-like
) ( non
empty
V106
()
V131
()
V132
()
V133
()
V134
()
V135
()
V136
()
V137
()
V141
()
V142
()
RealNormSpace-like
)
RealNormSpace
) : ( ( ) ( non
empty
)
set
)
:]
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) : ( ( ) ( )
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) &
f1
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,)
is_continuous_in
x0
: ( (
real
) (
V11
()
real
ext-real
)
number
) &
z
: ( ( ) ( )
Point
of ( ( ) ( non
empty
)
set
) )
=
f1
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,)
/.
x0
: ( (
real
) (
V11
()
real
ext-real
)
number
) : ( ( ) (
Relation-like
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) )
-defined
Function-like
V33
()
V34
()
V35
()
finite
V71
(
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
FinSubsequence-like
)
Element
of
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) )) ) &
f2
: ( (
Function-like
) (
Relation-like
the
carrier
of
(
REAL-NS
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( non
empty
strict
) ( non
empty
V106
()
V131
()
V132
()
V133
()
V134
()
V135
()
V136
()
V137
()
V141
()
V142
()
strict
RealNormSpace-like
V174
() )
NORMSTR
) : ( ( ) ( non
empty
)
set
)
-defined
the
carrier
of
b
3
: ( ( non
empty
V106
()
V131
()
V132
()
V133
()
V134
()
V135
()
V136
()
V137
()
V141
()
V142
()
RealNormSpace-like
) ( non
empty
V106
()
V131
()
V132
()
V133
()
V134
()
V135
()
V136
()
V137
()
V141
()
V142
()
RealNormSpace-like
)
RealNormSpace
) : ( ( ) ( non
empty
)
set
)
-valued
Function-like
)
PartFunc
of ,)
is_continuous_in
z
: ( ( ) ( )
Point
of ( ( ) ( non
empty
)
set
) ) holds
f2
: ( (
Function-like
) (
Relation-like
the
carrier
of
(
REAL-NS
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( non
empty
strict
) ( non
empty
V106
()
V131
()
V132
()
V133
()
V134
()
V135
()
V136
()
V137
()
V141
()
V142
()
strict
RealNormSpace-like
V174
() )
NORMSTR
) : ( ( ) ( non
empty
)
set
)
-defined
the
carrier
of
b
3
: ( ( non
empty
V106
()
V131
()
V132
()
V133
()
V134
()
V135
()
V136
()
V137
()
V141
()
V142
()
RealNormSpace-like
) ( non
empty
V106
()
V131
()
V132
()
V133
()
V134
()
V135
()
V136
()
V137
()
V141
()
V142
()
RealNormSpace-like
)
RealNormSpace
) : ( ( ) ( non
empty
)
set
)
-valued
Function-like
)
PartFunc
of ,)
*
f1
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,) : ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
the
carrier
of
b
3
: ( ( non
empty
V106
()
V131
()
V132
()
V133
()
V134
()
V135
()
V136
()
V137
()
V141
()
V142
()
RealNormSpace-like
) ( non
empty
V106
()
V131
()
V132
()
V133
()
V134
()
V135
()
V136
()
V137
()
V141
()
V142
()
RealNormSpace-like
)
RealNormSpace
) : ( ( ) ( non
empty
)
set
)
-valued
Function-like
)
Element
of
bool
[:
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) , the
carrier
of
b
3
: ( ( non
empty
V106
()
V131
()
V132
()
V133
()
V134
()
V135
()
V136
()
V137
()
V141
()
V142
()
RealNormSpace-like
) ( non
empty
V106
()
V131
()
V132
()
V133
()
V134
()
V135
()
V136
()
V137
()
V141
()
V142
()
RealNormSpace-like
)
RealNormSpace
) : ( ( ) ( non
empty
)
set
)
:]
: ( ( ) ( )
set
) : ( ( ) ( )
set
) )
is_continuous_in
x0
: ( (
real
) (
V11
()
real
ext-real
)
number
) ;
theorem
:: NFCONT_4:20
for
x0
being ( (
real
) (
V11
()
real
ext-real
)
number
)
for
S
being ( ( non
empty
V106
()
V131
()
V132
()
V133
()
V134
()
V135
()
V136
()
V137
()
V141
()
V142
()
RealNormSpace-like
) ( non
empty
V106
()
V131
()
V132
()
V133
()
V134
()
V135
()
V136
()
V137
()
V141
()
V142
()
RealNormSpace-like
)
RealNormSpace
)
for
f1
being ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
the
carrier
of
b
2
: ( ( non
empty
V106
()
V131
()
V132
()
V133
()
V134
()
V135
()
V136
()
V137
()
V141
()
V142
()
RealNormSpace-like
) ( non
empty
V106
()
V131
()
V132
()
V133
()
V134
()
V135
()
V136
()
V137
()
V141
()
V142
()
RealNormSpace-like
)
RealNormSpace
) : ( ( ) ( non
empty
)
set
)
-valued
Function-like
)
PartFunc
of ,)
for
f2
being ( (
Function-like
) (
Relation-like
the
carrier
of
b
2
: ( ( non
empty
V106
()
V131
()
V132
()
V133
()
V134
()
V135
()
V136
()
V137
()
V141
()
V142
()
RealNormSpace-like
) ( non
empty
V106
()
V131
()
V132
()
V133
()
V134
()
V135
()
V136
()
V137
()
V141
()
V142
()
RealNormSpace-like
)
RealNormSpace
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-valued
Function-like
V33
()
V34
()
V35
() )
PartFunc
of ,) st
x0
: ( (
real
) (
V11
()
real
ext-real
)
number
)
in
dom
(
f2
: ( (
Function-like
) (
Relation-like
the
carrier
of
b
2
: ( ( non
empty
V106
()
V131
()
V132
()
V133
()
V134
()
V135
()
V136
()
V137
()
V141
()
V142
()
RealNormSpace-like
) ( non
empty
V106
()
V131
()
V132
()
V133
()
V134
()
V135
()
V136
()
V137
()
V141
()
V142
()
RealNormSpace-like
)
RealNormSpace
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-valued
Function-like
V33
()
V34
()
V35
() )
PartFunc
of ,)
*
f1
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
the
carrier
of
b
2
: ( ( non
empty
V106
()
V131
()
V132
()
V133
()
V134
()
V135
()
V136
()
V137
()
V141
()
V142
()
RealNormSpace-like
) ( non
empty
V106
()
V131
()
V132
()
V133
()
V134
()
V135
()
V136
()
V137
()
V141
()
V142
()
RealNormSpace-like
)
RealNormSpace
) : ( ( ) ( non
empty
)
set
)
-valued
Function-like
)
PartFunc
of ,)
)
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-valued
Function-like
V33
()
V34
()
V35
() )
Element
of
bool
[:
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ,
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
:]
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) : ( ( ) ( )
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) &
f1
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
the
carrier
of
b
2
: ( ( non
empty
V106
()
V131
()
V132
()
V133
()
V134
()
V135
()
V136
()
V137
()
V141
()
V142
()
RealNormSpace-like
) ( non
empty
V106
()
V131
()
V132
()
V133
()
V134
()
V135
()
V136
()
V137
()
V141
()
V142
()
RealNormSpace-like
)
RealNormSpace
) : ( ( ) ( non
empty
)
set
)
-valued
Function-like
)
PartFunc
of ,)
is_continuous_in
x0
: ( (
real
) (
V11
()
real
ext-real
)
number
) &
f2
: ( (
Function-like
) (
Relation-like
the
carrier
of
b
2
: ( ( non
empty
V106
()
V131
()
V132
()
V133
()
V134
()
V135
()
V136
()
V137
()
V141
()
V142
()
RealNormSpace-like
) ( non
empty
V106
()
V131
()
V132
()
V133
()
V134
()
V135
()
V136
()
V137
()
V141
()
V142
()
RealNormSpace-like
)
RealNormSpace
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-valued
Function-like
V33
()
V34
()
V35
() )
PartFunc
of ,)
is_continuous_in
f1
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
the
carrier
of
b
2
: ( ( non
empty
V106
()
V131
()
V132
()
V133
()
V134
()
V135
()
V136
()
V137
()
V141
()
V142
()
RealNormSpace-like
) ( non
empty
V106
()
V131
()
V132
()
V133
()
V134
()
V135
()
V136
()
V137
()
V141
()
V142
()
RealNormSpace-like
)
RealNormSpace
) : ( ( ) ( non
empty
)
set
)
-valued
Function-like
)
PartFunc
of ,)
/.
x0
: ( (
real
) (
V11
()
real
ext-real
)
number
) : ( ( ) ( )
Element
of the
carrier
of
b
2
: ( ( non
empty
V106
()
V131
()
V132
()
V133
()
V134
()
V135
()
V136
()
V137
()
V141
()
V142
()
RealNormSpace-like
) ( non
empty
V106
()
V131
()
V132
()
V133
()
V134
()
V135
()
V136
()
V137
()
V141
()
V142
()
RealNormSpace-like
)
RealNormSpace
) : ( ( ) ( non
empty
)
set
) ) holds
f2
: ( (
Function-like
) (
Relation-like
the
carrier
of
b
2
: ( ( non
empty
V106
()
V131
()
V132
()
V133
()
V134
()
V135
()
V136
()
V137
()
V141
()
V142
()
RealNormSpace-like
) ( non
empty
V106
()
V131
()
V132
()
V133
()
V134
()
V135
()
V136
()
V137
()
V141
()
V142
()
RealNormSpace-like
)
RealNormSpace
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-valued
Function-like
V33
()
V34
()
V35
() )
PartFunc
of ,)
*
f1
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
the
carrier
of
b
2
: ( ( non
empty
V106
()
V131
()
V132
()
V133
()
V134
()
V135
()
V136
()
V137
()
V141
()
V142
()
RealNormSpace-like
) ( non
empty
V106
()
V131
()
V132
()
V133
()
V134
()
V135
()
V136
()
V137
()
V141
()
V142
()
RealNormSpace-like
)
RealNormSpace
) : ( ( ) ( non
empty
)
set
)
-valued
Function-like
)
PartFunc
of ,) : ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-valued
Function-like
V33
()
V34
()
V35
() )
Element
of
bool
[:
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ,
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
:]
: ( ( ) ( )
set
) : ( ( ) ( )
set
) )
is_continuous_in
x0
: ( (
real
) (
V11
()
real
ext-real
)
number
) ;
definition
let
n
be ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) ;
let
f
be ( (
Function-like
) (
Relation-like
REAL
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-defined
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-valued
Function-like
V33
()
V34
()
V35
() )
PartFunc
of ,) ;
let
x0
be ( ( ) (
Relation-like
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) )
-defined
Function-like
V33
()
V34
()
V35
()
finite
V71
(
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
FinSubsequence-like
)
Element
of
REAL
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) )) ) ;
pred
f
is_continuous_in
x0
means
:: NFCONT_4:def 4
ex
y0
being ( ( ) ( )
Point
of ( ( ) ( non
empty
)
set
) ) ex
g
being ( (
Function-like
) (
Relation-like
the
carrier
of
(
REAL-NS
n
: ( ( ) ( )
NORMSTR
)
)
: ( ( non
empty
strict
) ( non
empty
strict
)
NORMSTR
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-valued
Function-like
V33
()
V34
()
V35
() )
PartFunc
of ,) st
(
x0
: ( (
Function-like
quasi_total
) (
Relation-like
[:
n
: ( ( ) ( )
NORMSTR
) ,
n
: ( ( ) ( )
NORMSTR
)
:]
: ( ( ) ( )
set
)
-defined
n
: ( ( ) ( )
NORMSTR
)
-valued
Function-like
quasi_total
)
Element
of
bool
[:
[:
n
: ( ( ) ( )
NORMSTR
) ,
n
: ( ( ) ( )
NORMSTR
)
:]
: ( ( ) ( )
set
) ,
n
: ( ( ) ( )
NORMSTR
)
:]
: ( ( ) ( )
set
) : ( ( ) ( )
set
) )
=
y0
: ( ( ) ( )
Point
of ( ( ) ( non
empty
)
set
) ) &
f
: ( ( ) ( )
Element
of
n
: ( ( ) ( )
NORMSTR
) )
=
g
: ( (
Function-like
) (
Relation-like
the
carrier
of
(
REAL-NS
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( non
empty
strict
) ( non
empty
V106
()
V131
()
V132
()
V133
()
V134
()
V135
()
V136
()
V137
()
V141
()
V142
()
strict
RealNormSpace-like
V174
() )
NORMSTR
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-valued
Function-like
V33
()
V34
()
V35
() )
PartFunc
of ,) &
g
: ( (
Function-like
) (
Relation-like
the
carrier
of
(
REAL-NS
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( non
empty
strict
) ( non
empty
V106
()
V131
()
V132
()
V133
()
V134
()
V135
()
V136
()
V137
()
V141
()
V142
()
strict
RealNormSpace-like
V174
() )
NORMSTR
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-valued
Function-like
V33
()
V34
()
V35
() )
PartFunc
of ,)
is_continuous_in
y0
: ( ( ) ( )
Point
of ( ( ) ( non
empty
)
set
) ) );
end;
theorem
:: NFCONT_4:21
for
n
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
for
f
being ( (
Function-like
) (
Relation-like
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-defined
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-valued
Function-like
V33
()
V34
()
V35
() )
PartFunc
of ,)
for
h
being ( (
Function-like
) (
Relation-like
the
carrier
of
(
REAL-NS
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( non
empty
strict
) ( non
empty
V106
()
V131
()
V132
()
V133
()
V134
()
V135
()
V136
()
V137
()
V141
()
V142
()
strict
RealNormSpace-like
V174
() )
NORMSTR
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-valued
Function-like
V33
()
V34
()
V35
() )
PartFunc
of ,)
for
x0
being ( ( ) (
Relation-like
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) )
-defined
Function-like
V33
()
V34
()
V35
()
finite
V71
(
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
FinSubsequence-like
)
Element
of
REAL
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) )) )
for
y0
being ( ( ) ( )
Point
of ( ( ) ( non
empty
)
set
) ) st
f
: ( (
Function-like
) (
Relation-like
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-defined
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-valued
Function-like
V33
()
V34
()
V35
() )
PartFunc
of ,)
=
h
: ( (
Function-like
) (
Relation-like
the
carrier
of
(
REAL-NS
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( non
empty
strict
) ( non
empty
V106
()
V131
()
V132
()
V133
()
V134
()
V135
()
V136
()
V137
()
V141
()
V142
()
strict
RealNormSpace-like
V174
() )
NORMSTR
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-valued
Function-like
V33
()
V34
()
V35
() )
PartFunc
of ,) &
x0
: ( ( ) (
Relation-like
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) )
-defined
Function-like
V33
()
V34
()
V35
()
finite
V71
(
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
FinSubsequence-like
)
Element
of
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) )) )
=
y0
: ( ( ) ( )
Point
of ( ( ) ( non
empty
)
set
) ) holds
(
f
: ( (
Function-like
) (
Relation-like
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-defined
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-valued
Function-like
V33
()
V34
()
V35
() )
PartFunc
of ,)
is_continuous_in
x0
: ( ( ) (
Relation-like
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) )
-defined
Function-like
V33
()
V34
()
V35
()
finite
V71
(
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
FinSubsequence-like
)
Element
of
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) )) ) iff
h
: ( (
Function-like
) (
Relation-like
the
carrier
of
(
REAL-NS
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( non
empty
strict
) ( non
empty
V106
()
V131
()
V132
()
V133
()
V134
()
V135
()
V136
()
V137
()
V141
()
V142
()
strict
RealNormSpace-like
V174
() )
NORMSTR
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-valued
Function-like
V33
()
V34
()
V35
() )
PartFunc
of ,)
is_continuous_in
y0
: ( ( ) ( )
Point
of ( ( ) ( non
empty
)
set
) ) ) ;
theorem
:: NFCONT_4:22
for
n
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
for
x0
being ( (
real
) (
V11
()
real
ext-real
)
number
)
for
f1
being ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,)
for
f2
being ( (
Function-like
) (
Relation-like
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-defined
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-valued
Function-like
V33
()
V34
()
V35
() )
PartFunc
of ,) st
x0
: ( (
real
) (
V11
()
real
ext-real
)
number
)
in
dom
(
f2
: ( (
Function-like
) (
Relation-like
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-defined
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-valued
Function-like
V33
()
V34
()
V35
() )
PartFunc
of ,)
*
f1
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,)
)
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-valued
Function-like
V33
()
V34
()
V35
() )
Element
of
bool
[:
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ,
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
:]
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) : ( ( ) ( )
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) &
f1
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,)
is_continuous_in
x0
: ( (
real
) (
V11
()
real
ext-real
)
number
) &
f2
: ( (
Function-like
) (
Relation-like
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-defined
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-valued
Function-like
V33
()
V34
()
V35
() )
PartFunc
of ,)
is_continuous_in
f1
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,)
/.
x0
: ( (
real
) (
V11
()
real
ext-real
)
number
) : ( ( ) (
Relation-like
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) )
-defined
Function-like
V33
()
V34
()
V35
()
finite
V71
(
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
FinSubsequence-like
)
Element
of
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) )) ) holds
f2
: ( (
Function-like
) (
Relation-like
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-defined
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-valued
Function-like
V33
()
V34
()
V35
() )
PartFunc
of ,)
*
f1
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,) : ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-valued
Function-like
V33
()
V34
()
V35
() )
Element
of
bool
[:
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ,
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
:]
: ( ( ) ( )
set
) : ( ( ) ( )
set
) )
is_continuous_in
x0
: ( (
real
) (
V11
()
real
ext-real
)
number
) ;
definition
let
n
be ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) ;
let
f
be ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,) ;
attr
f
is
continuous
means
:: NFCONT_4:def 5
for
x0
being ( (
real
) (
V11
()
real
ext-real
)
number
) st
x0
: ( (
real
) (
V11
()
real
ext-real
)
number
)
in
dom
f
: ( ( ) ( )
Element
of
n
: ( ( ) ( )
NORMSTR
) ) : ( ( ) ( )
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) holds
f
: ( ( ) ( )
Element
of
n
: ( ( ) ( )
NORMSTR
) )
is_continuous_in
x0
: ( (
real
) (
V11
()
real
ext-real
)
number
) ;
end;
theorem
:: NFCONT_4:23
for
n
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
for
g
being ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
the
carrier
of
(
REAL-NS
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( non
empty
strict
) ( non
empty
V106
()
V131
()
V132
()
V133
()
V134
()
V135
()
V136
()
V137
()
V141
()
V142
()
strict
RealNormSpace-like
V174
() )
NORMSTR
) : ( ( ) ( non
empty
)
set
)
-valued
Function-like
)
PartFunc
of ,)
for
f
being ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,) st
g
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
the
carrier
of
(
REAL-NS
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( non
empty
strict
) ( non
empty
V106
()
V131
()
V132
()
V133
()
V134
()
V135
()
V136
()
V137
()
V141
()
V142
()
strict
RealNormSpace-like
V174
() )
NORMSTR
) : ( ( ) ( non
empty
)
set
)
-valued
Function-like
)
PartFunc
of ,)
=
f
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,) holds
(
g
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
the
carrier
of
(
REAL-NS
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( non
empty
strict
) ( non
empty
V106
()
V131
()
V132
()
V133
()
V134
()
V135
()
V136
()
V137
()
V141
()
V142
()
strict
RealNormSpace-like
V174
() )
NORMSTR
) : ( ( ) ( non
empty
)
set
)
-valued
Function-like
)
PartFunc
of ,) is
continuous
iff
f
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,) is
continuous
) ;
theorem
:: NFCONT_4:24
for
n
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
for
X
being ( ( ) ( )
set
)
for
f
being ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,) st
X
: ( ( ) ( )
set
)
c=
dom
f
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,) : ( ( ) ( )
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) holds
(
f
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,)
|
X
: ( ( ) ( )
set
) : ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
Element
of
bool
[:
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ,
(
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
:]
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) is
continuous
iff for
x0
being ( (
real
) (
V11
()
real
ext-real
)
number
)
for
r
being ( ( ) (
V11
()
real
ext-real
)
Real
) st
x0
: ( (
real
) (
V11
()
real
ext-real
)
number
)
in
X
: ( ( ) ( )
set
) &
0
: ( ( ) (
empty
trivial
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V11
()
real
ext-real
non
positive
non
negative
Function-like
functional
FinSequence-membered
V227
()
V228
()
V229
()
V230
()
V231
()
V232
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
<
r
: ( ( ) (
V11
()
real
ext-real
)
Real
) holds
ex
s
being ( (
real
) (
V11
()
real
ext-real
)
number
) st
(
0
: ( ( ) (
empty
trivial
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V11
()
real
ext-real
non
positive
non
negative
Function-like
functional
FinSequence-membered
V227
()
V228
()
V229
()
V230
()
V231
()
V232
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
<
s
: ( (
real
) (
V11
()
real
ext-real
)
number
) & ( for
x1
being ( (
real
) (
V11
()
real
ext-real
)
number
) st
x1
: ( (
real
) (
V11
()
real
ext-real
)
number
)
in
X
: ( ( ) ( )
set
) &
abs
(
x1
: ( (
real
) (
V11
()
real
ext-real
)
number
)
-
x0
: ( (
real
) (
V11
()
real
ext-real
)
number
)
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) )
<
s
: ( (
real
) (
V11
()
real
ext-real
)
number
) holds
|.
(
(
f
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,)
/.
x1
: ( (
real
) (
V11
()
real
ext-real
)
number
)
)
: ( ( ) (
Relation-like
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) )
-defined
Function-like
V33
()
V34
()
V35
()
finite
V71
(
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
FinSubsequence-like
)
Element
of
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) )) )
-
(
f
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,)
/.
x0
: ( (
real
) (
V11
()
real
ext-real
)
number
)
)
: ( ( ) (
Relation-like
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) )
-defined
Function-like
V33
()
V34
()
V35
()
finite
V71
(
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
FinSubsequence-like
)
Element
of
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) )) )
)
: ( ( ) (
Relation-like
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) )
-defined
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-valued
Function-like
V33
()
V34
()
V35
()
finite
V71
(
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
FinSubsequence-like
)
M13
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ,
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) )) ))
.|
: ( ( ) (
V11
()
real
ext-real
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) )
<
r
: ( ( ) (
V11
()
real
ext-real
)
Real
) ) ) ) ;
registration
let
n
be ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) ;
cluster
Function-like
constant
->
Function-like
continuous
for ( ( ) ( )
Element
of
bool
[:
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ,
(
REAL
n
: ( ( ) ( )
NORMSTR
)
)
: ( ( ) ( )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
:]
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ;
end;
registration
let
n
be ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) ;
cluster
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
()
continuous
for ( ( ) ( )
Element
of
bool
[:
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ,
(
REAL
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
:]
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ;
end;
registration
let
n
be ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) ;
let
f
be ( (
Function-like
continuous
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
()
continuous
)
PartFunc
of ,) ;
let
X
be ( ( ) ( )
set
) ;
cluster
f
: ( (
Function-like
continuous
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
()
continuous
)
Element
of
bool
[:
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ,
(
REAL
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
:]
: ( ( ) ( )
set
) : ( ( ) ( )
set
) )
|
X
: ( ( ) ( )
set
) : ( (
Relation-like
) (
Relation-like
Function-like
)
set
)
->
Function-like
continuous
for ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,) ;
end;
theorem
:: NFCONT_4:25
for
n
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
for
X
,
X1
being ( ( ) ( )
set
)
for
f
being ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,) st
f
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,)
|
X
: ( ( ) ( )
set
) : ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
Element
of
bool
[:
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ,
(
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
:]
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) is
continuous
&
X1
: ( ( ) ( )
set
)
c=
X
: ( ( ) ( )
set
) holds
f
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,)
|
X1
: ( ( ) ( )
set
) : ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
Element
of
bool
[:
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ,
(
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
:]
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) is
continuous
;
registration
let
n
be ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) ;
cluster
empty
Function-like
->
Function-like
continuous
for ( ( ) ( )
Element
of
bool
[:
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ,
(
REAL
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
:]
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ;
end;
registration
let
n
be ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) ;
let
f
be ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,) ;
let
X
be ( (
trivial
) (
trivial
)
set
) ;
cluster
f
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
Element
of
bool
[:
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ,
(
REAL
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
:]
: ( ( ) ( )
set
) : ( ( ) ( )
set
) )
|
X
: ( (
trivial
) (
trivial
)
set
) : ( (
Relation-like
) (
Relation-like
Function-like
)
set
)
->
Function-like
continuous
for ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,) ;
end;
registration
let
n
be ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) ;
let
f1
,
f2
be ( (
Function-like
continuous
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
()
continuous
)
PartFunc
of ,) ;
cluster
f1
: ( (
Function-like
continuous
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
()
continuous
)
Element
of
bool
[:
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ,
(
REAL
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
:]
: ( ( ) ( )
set
) : ( ( ) ( )
set
) )
+
f2
: ( (
Function-like
continuous
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
()
continuous
)
Element
of
bool
[:
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ,
(
REAL
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
:]
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) : ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
Element
of
bool
[:
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ,
(
REAL
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
:]
: ( ( ) ( )
set
) : ( ( ) ( )
set
) )
->
Function-like
continuous
for ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,) ;
end;
theorem
:: NFCONT_4:26
for
n
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
for
X
being ( ( ) ( )
set
)
for
f1
,
f2
being ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,) st
X
: ( ( ) ( )
set
)
c=
(
dom
f1
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,)
)
: ( ( ) ( )
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) )
/\
(
dom
f2
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,)
)
: ( ( ) ( )
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) : ( ( ) ( )
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) &
f1
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,)
|
X
: ( ( ) ( )
set
) : ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
Element
of
bool
[:
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ,
(
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
:]
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) is
continuous
&
f2
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,)
|
X
: ( ( ) ( )
set
) : ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
Element
of
bool
[:
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ,
(
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
:]
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) is
continuous
holds
(
(
f1
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,)
+
f2
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,)
)
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
Element
of
bool
[:
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ,
(
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
:]
: ( ( ) ( )
set
) : ( ( ) ( )
set
) )
|
X
: ( ( ) ( )
set
) : ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
Element
of
bool
[:
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ,
(
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
:]
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) is
continuous
&
(
f1
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,)
-
f2
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,)
)
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
Element
of
bool
[:
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ,
(
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
:]
: ( ( ) ( )
set
) : ( ( ) ( )
set
) )
|
X
: ( ( ) ( )
set
) : ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
Element
of
bool
[:
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ,
(
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
:]
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) is
continuous
) ;
theorem
:: NFCONT_4:27
for
n
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
for
X
,
X1
being ( ( ) ( )
set
)
for
f1
,
f2
being ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,) st
X
: ( ( ) ( )
set
)
c=
dom
f1
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,) : ( ( ) ( )
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) &
X1
: ( ( ) ( )
set
)
c=
dom
f2
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,) : ( ( ) ( )
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) &
f1
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,)
|
X
: ( ( ) ( )
set
) : ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
Element
of
bool
[:
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ,
(
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
:]
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) is
continuous
&
f2
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,)
|
X1
: ( ( ) ( )
set
) : ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
Element
of
bool
[:
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ,
(
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
:]
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) is
continuous
holds
(
(
f1
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,)
+
f2
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,)
)
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
Element
of
bool
[:
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ,
(
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
:]
: ( ( ) ( )
set
) : ( ( ) ( )
set
) )
|
(
X
: ( ( ) ( )
set
)
/\
X1
: ( ( ) ( )
set
)
)
: ( ( ) ( )
set
) : ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
Element
of
bool
[:
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ,
(
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
:]
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) is
continuous
&
(
f1
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,)
-
f2
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,)
)
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
Element
of
bool
[:
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ,
(
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
:]
: ( ( ) ( )
set
) : ( ( ) ( )
set
) )
|
(
X
: ( ( ) ( )
set
)
/\
X1
: ( ( ) ( )
set
)
)
: ( ( ) ( )
set
) : ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
Element
of
bool
[:
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ,
(
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
:]
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) is
continuous
) ;
registration
let
n
be ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) ;
let
f
be ( (
Function-like
continuous
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
()
continuous
)
PartFunc
of ,) ;
let
r
be ( ( ) (
V11
()
real
ext-real
)
Real
) ;
cluster
r
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) )
(#)
f
: ( (
Function-like
continuous
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
()
continuous
)
Element
of
bool
[:
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ,
(
REAL
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
:]
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) : ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
Element
of
bool
[:
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ,
(
REAL
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
:]
: ( ( ) ( )
set
) : ( ( ) ( )
set
) )
->
Function-like
continuous
for ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,) ;
end;
theorem
:: NFCONT_4:28
for
n
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
for
X
being ( ( ) ( )
set
)
for
r
being ( ( ) (
V11
()
real
ext-real
)
Real
)
for
f
being ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,) st
X
: ( ( ) ( )
set
)
c=
dom
f
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,) : ( ( ) ( )
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) &
f
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,)
|
X
: ( ( ) ( )
set
) : ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
Element
of
bool
[:
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ,
(
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
:]
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) is
continuous
holds
(
r
: ( ( ) (
V11
()
real
ext-real
)
Real
)
(#)
f
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,)
)
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
Element
of
bool
[:
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ,
(
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
:]
: ( ( ) ( )
set
) : ( ( ) ( )
set
) )
|
X
: ( ( ) ( )
set
) : ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
Element
of
bool
[:
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ,
(
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
:]
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) is
continuous
;
theorem
:: NFCONT_4:29
for
n
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
for
X
being ( ( ) ( )
set
)
for
f
being ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,) st
X
: ( ( ) ( )
set
)
c=
dom
f
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,) : ( ( ) ( )
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) &
f
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,)
|
X
: ( ( ) ( )
set
) : ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
Element
of
bool
[:
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ,
(
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
:]
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) is
continuous
holds
(
|.
f
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,)
.|
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-valued
Function-like
V33
()
V34
()
V35
() )
PartFunc
of ,)
|
X
: ( ( ) ( )
set
) : ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-valued
Function-like
V33
()
V34
()
V35
() )
Element
of
bool
[:
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ,
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
:]
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) is
continuous
&
(
-
f
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,)
)
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,)
|
X
: ( ( ) ( )
set
) : ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
Element
of
bool
[:
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ,
(
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
:]
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) is
continuous
) ;
theorem
:: NFCONT_4:30
for
n
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
for
f
being ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,) st
f
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,) is
total
& ( for
x1
,
x2
being ( (
real
) (
V11
()
real
ext-real
)
number
) holds
f
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,)
/.
(
x1
: ( (
real
) (
V11
()
real
ext-real
)
number
)
+
x2
: ( (
real
) (
V11
()
real
ext-real
)
number
)
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ) : ( ( ) (
Relation-like
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) )
-defined
Function-like
V33
()
V34
()
V35
()
finite
V71
(
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
FinSubsequence-like
)
Element
of
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) )) )
=
(
f
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,)
/.
x1
: ( (
real
) (
V11
()
real
ext-real
)
number
)
)
: ( ( ) (
Relation-like
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) )
-defined
Function-like
V33
()
V34
()
V35
()
finite
V71
(
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
FinSubsequence-like
)
Element
of
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) )) )
+
(
f
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,)
/.
x2
: ( (
real
) (
V11
()
real
ext-real
)
number
)
)
: ( ( ) (
Relation-like
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) )
-defined
Function-like
V33
()
V34
()
V35
()
finite
V71
(
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
FinSubsequence-like
)
Element
of
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) )) ) : ( ( ) (
Relation-like
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) )
-defined
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-valued
Function-like
V33
()
V34
()
V35
()
finite
V71
(
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
FinSubsequence-like
)
M13
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ,
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) )) )) ) & ex
x0
being ( (
real
) (
V11
()
real
ext-real
)
number
) st
f
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,)
is_continuous_in
x0
: ( (
real
) (
V11
()
real
ext-real
)
number
) holds
f
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,)
|
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
Element
of
bool
[:
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ,
(
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
:]
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) is
continuous
;
theorem
:: NFCONT_4:31
for
n
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
for
f
being ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,)
for
Y
being ( ( ) ( )
Subset
of ) st
dom
f
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,) : ( ( ) ( )
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) is
compact
&
f
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,)
|
(
dom
f
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,)
)
: ( ( ) ( )
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) : ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
Element
of
bool
[:
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ,
(
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
:]
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) is
continuous
&
Y
: ( ( ) ( )
Subset
of )
=
rng
f
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,) : ( ( ) (
functional
FinSequence-membered
V227
()
V228
()
V229
() )
Element
of
bool
(
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) )) : ( ( ) ( )
set
) ) holds
Y
: ( ( ) ( )
Subset
of ) is
compact
;
theorem
:: NFCONT_4:32
for
n
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
for
f
being ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,)
for
Y
being ( ( ) ( )
Subset
of ( ( ) ( )
set
) )
for
Z
being ( ( ) ( )
Subset
of ) st
Y
: ( ( ) ( )
Subset
of ( ( ) ( )
set
) )
c=
dom
f
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,) : ( ( ) ( )
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) &
Z
: ( ( ) ( )
Subset
of )
=
f
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,)
.:
Y
: ( ( ) ( )
Subset
of ( ( ) ( )
set
) ) : ( ( ) (
functional
FinSequence-membered
V227
()
V228
()
V229
() )
Element
of
bool
(
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) )) : ( ( ) ( )
set
) ) &
Y
: ( ( ) ( )
Subset
of ( ( ) ( )
set
) ) is
compact
&
f
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,)
|
Y
: ( ( ) ( )
Subset
of ( ( ) ( )
set
) ) : ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
Element
of
bool
[:
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ,
(
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
:]
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) is
continuous
holds
Z
: ( ( ) ( )
Subset
of ) is
compact
;
definition
let
n
be ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) ;
let
f
be ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,) ;
attr
f
is
Lipschitzian
means
:: NFCONT_4:def 6
ex
g
being ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
the
carrier
of
(
REAL-NS
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( non
empty
strict
) ( non
empty
V106
()
V131
()
V132
()
V133
()
V134
()
V135
()
V136
()
V137
()
V141
()
V142
()
strict
RealNormSpace-like
V174
() )
NORMSTR
) : ( ( ) ( non
empty
)
set
)
-valued
Function-like
)
PartFunc
of ,) st
(
g
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
the
carrier
of
(
REAL-NS
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( non
empty
strict
) ( non
empty
V106
()
V131
()
V132
()
V133
()
V134
()
V135
()
V136
()
V137
()
V141
()
V142
()
strict
RealNormSpace-like
V174
() )
NORMSTR
) : ( ( ) ( non
empty
)
set
)
-valued
Function-like
)
PartFunc
of ,)
=
f
: ( (
Relation-like
Function-like
) (
Relation-like
Function-like
)
set
) &
g
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
the
carrier
of
(
REAL-NS
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( non
empty
strict
) ( non
empty
V106
()
V131
()
V132
()
V133
()
V134
()
V135
()
V136
()
V137
()
V141
()
V142
()
strict
RealNormSpace-like
V174
() )
NORMSTR
) : ( ( ) ( non
empty
)
set
)
-valued
Function-like
)
PartFunc
of ,) is
Lipschitzian
);
end;
theorem
:: NFCONT_4:33
for
n
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
for
f
being ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,) holds
(
f
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,) is
Lipschitzian
iff ex
r
being ( (
real
) (
V11
()
real
ext-real
)
number
) st
(
0
: ( ( ) (
empty
trivial
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V11
()
real
ext-real
non
positive
non
negative
Function-like
functional
FinSequence-membered
V227
()
V228
()
V229
()
V230
()
V231
()
V232
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
<
r
: ( (
real
) (
V11
()
real
ext-real
)
number
) & ( for
x1
,
x2
being ( (
real
) (
V11
()
real
ext-real
)
number
) st
x1
: ( (
real
) (
V11
()
real
ext-real
)
number
)
in
dom
f
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,) : ( ( ) ( )
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) &
x2
: ( (
real
) (
V11
()
real
ext-real
)
number
)
in
dom
f
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,) : ( ( ) ( )
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) holds
|.
(
(
f
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,)
/.
x1
: ( (
real
) (
V11
()
real
ext-real
)
number
)
)
: ( ( ) (
Relation-like
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) )
-defined
Function-like
V33
()
V34
()
V35
()
finite
V71
(
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
FinSubsequence-like
)
Element
of
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) )) )
-
(
f
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,)
/.
x2
: ( (
real
) (
V11
()
real
ext-real
)
number
)
)
: ( ( ) (
Relation-like
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) )
-defined
Function-like
V33
()
V34
()
V35
()
finite
V71
(
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
FinSubsequence-like
)
Element
of
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) )) )
)
: ( ( ) (
Relation-like
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) )
-defined
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-valued
Function-like
V33
()
V34
()
V35
()
finite
V71
(
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
FinSubsequence-like
)
M13
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ,
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) )) ))
.|
: ( ( ) (
V11
()
real
ext-real
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) )
<=
r
: ( (
real
) (
V11
()
real
ext-real
)
number
)
*
(
abs
(
x1
: ( (
real
) (
V11
()
real
ext-real
)
number
)
-
x2
: ( (
real
) (
V11
()
real
ext-real
)
number
)
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) )
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ) ) ) ) ;
theorem
:: NFCONT_4:34
for
n
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
for
f
being ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,)
for
h
being ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
the
carrier
of
(
REAL-NS
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( non
empty
strict
) ( non
empty
V106
()
V131
()
V132
()
V133
()
V134
()
V135
()
V136
()
V137
()
V141
()
V142
()
strict
RealNormSpace-like
V174
() )
NORMSTR
) : ( ( ) ( non
empty
)
set
)
-valued
Function-like
)
PartFunc
of ,) st
f
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,)
=
h
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
the
carrier
of
(
REAL-NS
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( non
empty
strict
) ( non
empty
V106
()
V131
()
V132
()
V133
()
V134
()
V135
()
V136
()
V137
()
V141
()
V142
()
strict
RealNormSpace-like
V174
() )
NORMSTR
) : ( ( ) ( non
empty
)
set
)
-valued
Function-like
)
PartFunc
of ,) holds
(
f
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,) is
Lipschitzian
iff
h
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
the
carrier
of
(
REAL-NS
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( non
empty
strict
) ( non
empty
V106
()
V131
()
V132
()
V133
()
V134
()
V135
()
V136
()
V137
()
V141
()
V142
()
strict
RealNormSpace-like
V174
() )
NORMSTR
) : ( ( ) ( non
empty
)
set
)
-valued
Function-like
)
PartFunc
of ,) is
Lipschitzian
) ;
theorem
:: NFCONT_4:35
for
n
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
for
X
being ( ( ) ( )
set
)
for
f
being ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,) holds
(
f
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,)
|
X
: ( ( ) ( )
set
) : ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
Element
of
bool
[:
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ,
(
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
:]
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) is
Lipschitzian
iff ex
r
being ( (
real
) (
V11
()
real
ext-real
)
number
) st
(
0
: ( ( ) (
empty
trivial
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V11
()
real
ext-real
non
positive
non
negative
Function-like
functional
FinSequence-membered
V227
()
V228
()
V229
()
V230
()
V231
()
V232
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
<
r
: ( (
real
) (
V11
()
real
ext-real
)
number
) & ( for
x1
,
x2
being ( (
real
) (
V11
()
real
ext-real
)
number
) st
x1
: ( (
real
) (
V11
()
real
ext-real
)
number
)
in
dom
(
f
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,)
|
X
: ( ( ) ( )
set
)
)
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
Element
of
bool
[:
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ,
(
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
:]
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) : ( ( ) ( )
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) &
x2
: ( (
real
) (
V11
()
real
ext-real
)
number
)
in
dom
(
f
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,)
|
X
: ( ( ) ( )
set
)
)
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
Element
of
bool
[:
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ,
(
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
:]
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) : ( ( ) ( )
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) holds
|.
(
(
f
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,)
/.
x1
: ( (
real
) (
V11
()
real
ext-real
)
number
)
)
: ( ( ) (
Relation-like
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) )
-defined
Function-like
V33
()
V34
()
V35
()
finite
V71
(
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
FinSubsequence-like
)
Element
of
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) )) )
-
(
f
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,)
/.
x2
: ( (
real
) (
V11
()
real
ext-real
)
number
)
)
: ( ( ) (
Relation-like
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) )
-defined
Function-like
V33
()
V34
()
V35
()
finite
V71
(
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
FinSubsequence-like
)
Element
of
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) )) )
)
: ( ( ) (
Relation-like
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) )
-defined
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-valued
Function-like
V33
()
V34
()
V35
()
finite
V71
(
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
FinSubsequence-like
)
M13
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ,
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) )) ))
.|
: ( ( ) (
V11
()
real
ext-real
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) )
<=
r
: ( (
real
) (
V11
()
real
ext-real
)
number
)
*
(
abs
(
x1
: ( (
real
) (
V11
()
real
ext-real
)
number
)
-
x2
: ( (
real
) (
V11
()
real
ext-real
)
number
)
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) )
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ) ) ) ) ;
registration
let
n
be ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) ;
cluster
empty
Function-like
->
Function-like
Lipschitzian
for ( ( ) ( )
Element
of
bool
[:
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ,
(
REAL
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
:]
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ;
end;
registration
let
n
be ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) ;
cluster
empty
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() for ( ( ) ( )
Element
of
bool
[:
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ,
(
REAL
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
:]
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ;
end;
registration
let
n
be ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) ;
let
f
be ( (
Function-like
Lipschitzian
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
()
Lipschitzian
)
PartFunc
of ,) ;
let
X
be ( ( ) ( )
set
) ;
cluster
f
: ( (
Function-like
Lipschitzian
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
()
Lipschitzian
)
Element
of
bool
[:
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ,
(
REAL
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
:]
: ( ( ) ( )
set
) : ( ( ) ( )
set
) )
|
X
: ( ( ) ( )
set
) : ( (
Relation-like
) (
Relation-like
Function-like
)
set
)
->
Function-like
Lipschitzian
for ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,) ;
end;
theorem
:: NFCONT_4:36
for
n
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
for
X
,
X1
being ( ( ) ( )
set
)
for
f
being ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,) st
f
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,)
|
X
: ( ( ) ( )
set
) : ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
Element
of
bool
[:
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ,
(
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
:]
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) is
Lipschitzian
&
X1
: ( ( ) ( )
set
)
c=
X
: ( ( ) ( )
set
) holds
f
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,)
|
X1
: ( ( ) ( )
set
) : ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
Element
of
bool
[:
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ,
(
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
:]
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) is
Lipschitzian
;
registration
let
n
be ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) ;
let
f1
,
f2
be ( (
Function-like
Lipschitzian
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
()
Lipschitzian
)
PartFunc
of ,) ;
cluster
f1
: ( (
Function-like
Lipschitzian
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
()
Lipschitzian
)
Element
of
bool
[:
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ,
(
REAL
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
:]
: ( ( ) ( )
set
) : ( ( ) ( )
set
) )
+
f2
: ( (
Function-like
Lipschitzian
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
()
Lipschitzian
)
Element
of
bool
[:
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ,
(
REAL
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
:]
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) : ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
Element
of
bool
[:
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ,
(
REAL
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
:]
: ( ( ) ( )
set
) : ( ( ) ( )
set
) )
->
Function-like
Lipschitzian
for ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,) ;
cluster
f1
: ( (
Function-like
Lipschitzian
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
()
Lipschitzian
)
Element
of
bool
[:
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ,
(
REAL
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
:]
: ( ( ) ( )
set
) : ( ( ) ( )
set
) )
-
f2
: ( (
Function-like
Lipschitzian
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
()
Lipschitzian
)
Element
of
bool
[:
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ,
(
REAL
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
:]
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) : ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
Element
of
bool
[:
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ,
(
REAL
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
:]
: ( ( ) ( )
set
) : ( ( ) ( )
set
) )
->
Function-like
Lipschitzian
for ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,) ;
end;
theorem
:: NFCONT_4:37
for
n
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
for
X
,
X1
being ( ( ) ( )
set
)
for
f1
,
f2
being ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,) st
f1
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,)
|
X
: ( ( ) ( )
set
) : ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
Element
of
bool
[:
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ,
(
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
:]
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) is
Lipschitzian
&
f2
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,)
|
X1
: ( ( ) ( )
set
) : ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
Element
of
bool
[:
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ,
(
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
:]
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) is
Lipschitzian
holds
(
f1
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,)
+
f2
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,)
)
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
Element
of
bool
[:
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ,
(
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
:]
: ( ( ) ( )
set
) : ( ( ) ( )
set
) )
|
(
X
: ( ( ) ( )
set
)
/\
X1
: ( ( ) ( )
set
)
)
: ( ( ) ( )
set
) : ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
Element
of
bool
[:
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ,
(
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
:]
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) is
Lipschitzian
;
theorem
:: NFCONT_4:38
for
n
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
for
X
,
X1
being ( ( ) ( )
set
)
for
f1
,
f2
being ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,) st
f1
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,)
|
X
: ( ( ) ( )
set
) : ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
Element
of
bool
[:
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ,
(
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
:]
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) is
Lipschitzian
&
f2
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,)
|
X1
: ( ( ) ( )
set
) : ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
Element
of
bool
[:
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ,
(
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
:]
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) is
Lipschitzian
holds
(
f1
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,)
-
f2
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,)
)
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
Element
of
bool
[:
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ,
(
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
:]
: ( ( ) ( )
set
) : ( ( ) ( )
set
) )
|
(
X
: ( ( ) ( )
set
)
/\
X1
: ( ( ) ( )
set
)
)
: ( ( ) ( )
set
) : ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
Element
of
bool
[:
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ,
(
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
:]
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) is
Lipschitzian
;
registration
let
n
be ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) ;
let
f
be ( (
Function-like
Lipschitzian
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
()
Lipschitzian
)
PartFunc
of ,) ;
let
p
be ( ( ) (
V11
()
real
ext-real
)
Real
) ;
cluster
p
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) )
(#)
f
: ( (
Function-like
Lipschitzian
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
()
Lipschitzian
)
Element
of
bool
[:
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ,
(
REAL
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
:]
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) : ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
Element
of
bool
[:
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ,
(
REAL
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
:]
: ( ( ) ( )
set
) : ( ( ) ( )
set
) )
->
Function-like
Lipschitzian
for ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,) ;
end;
theorem
:: NFCONT_4:39
for
n
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
for
X
being ( ( ) ( )
set
)
for
p
being ( ( ) (
V11
()
real
ext-real
)
Real
)
for
f
being ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,) st
f
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,)
|
X
: ( ( ) ( )
set
) : ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
Element
of
bool
[:
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ,
(
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
:]
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) is
Lipschitzian
&
X
: ( ( ) ( )
set
)
c=
dom
f
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,) : ( ( ) ( )
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) holds
(
p
: ( ( ) (
V11
()
real
ext-real
)
Real
)
(#)
f
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,)
)
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
Element
of
bool
[:
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ,
(
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
:]
: ( ( ) ( )
set
) : ( ( ) ( )
set
) )
|
X
: ( ( ) ( )
set
) : ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
Element
of
bool
[:
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ,
(
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
:]
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) is
Lipschitzian
;
registration
let
n
be ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) ;
let
f
be ( (
Function-like
Lipschitzian
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
()
Lipschitzian
)
PartFunc
of ,) ;
cluster
|.
f
: ( (
Function-like
Lipschitzian
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
()
Lipschitzian
)
Element
of
bool
[:
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ,
(
REAL
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
:]
: ( ( ) ( )
set
) : ( ( ) ( )
set
) )
.|
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-valued
Function-like
V33
()
V34
()
V35
() )
PartFunc
of ,)
->
Function-like
Lipschitzian
for ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-valued
Function-like
V33
()
V34
()
V35
() )
PartFunc
of ,) ;
end;
theorem
:: NFCONT_4:40
for
n
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
for
X
being ( ( ) ( )
set
)
for
f
being ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,) st
f
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,)
|
X
: ( ( ) ( )
set
) : ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
Element
of
bool
[:
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ,
(
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
:]
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) is
Lipschitzian
holds
(
-
(
f
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,)
|
X
: ( ( ) ( )
set
)
)
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
Element
of
bool
[:
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ,
(
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
:]
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) : ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,) is
Lipschitzian
&
|.
f
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,)
.|
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-valued
Function-like
V33
()
V34
()
V35
() )
PartFunc
of ,)
|
X
: ( ( ) ( )
set
) : ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-valued
Function-like
V33
()
V34
()
V35
() )
Element
of
bool
[:
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ,
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
:]
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) is
Lipschitzian
&
(
-
f
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,)
)
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,)
|
X
: ( ( ) ( )
set
) : ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
Element
of
bool
[:
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ,
(
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
:]
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) is
Lipschitzian
) ;
registration
let
n
be ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) ;
cluster
Function-like
constant
->
Function-like
Lipschitzian
for ( ( ) ( )
Element
of
bool
[:
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ,
(
REAL
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
:]
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ;
end;
registration
let
n
be ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) ;
cluster
Function-like
Lipschitzian
->
Function-like
continuous
for ( ( ) ( )
Element
of
bool
[:
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ,
(
REAL
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
:]
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) ;
end;
theorem
:: NFCONT_4:41
for
n
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
for
X
being ( ( ) ( )
set
)
for
f
being ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,)
for
r
,
p
being ( ( ) (
Relation-like
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) )
-defined
Function-like
V33
()
V34
()
V35
()
finite
V71
(
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
FinSubsequence-like
)
Element
of
REAL
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) )) ) st ( for
x0
being ( (
real
) (
V11
()
real
ext-real
)
number
) st
x0
: ( (
real
) (
V11
()
real
ext-real
)
number
)
in
X
: ( ( ) ( )
set
) holds
f
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,)
/.
x0
: ( (
real
) (
V11
()
real
ext-real
)
number
) : ( ( ) (
Relation-like
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) )
-defined
Function-like
V33
()
V34
()
V35
()
finite
V71
(
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
FinSubsequence-like
)
Element
of
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) )) )
=
(
x0
: ( (
real
) (
V11
()
real
ext-real
)
number
)
*
r
: ( ( ) (
Relation-like
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) )
-defined
Function-like
V33
()
V34
()
V35
()
finite
V71
(
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
FinSubsequence-like
)
Element
of
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) )) )
)
: ( ( ) (
Relation-like
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) )
-defined
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-valued
Function-like
V33
()
V34
()
V35
()
finite
V71
(
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
FinSubsequence-like
)
M13
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ,
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) )) ))
+
p
: ( ( ) (
Relation-like
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) )
-defined
Function-like
V33
()
V34
()
V35
()
finite
V71
(
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
FinSubsequence-like
)
Element
of
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) )) ) : ( ( ) (
Relation-like
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) )
-defined
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-valued
Function-like
V33
()
V34
()
V35
()
finite
V71
(
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
FinSubsequence-like
)
M13
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ,
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) )) )) ) holds
f
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,)
|
X
: ( ( ) ( )
set
) : ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
Element
of
bool
[:
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ,
(
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
:]
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) is
continuous
;
theorem
:: NFCONT_4:42
for
n
,
i
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
for
x0
being ( ( ) (
Relation-like
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) )
-defined
Function-like
V33
()
V34
()
V35
()
finite
V71
(
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
FinSubsequence-like
)
Element
of
REAL
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) )) ) st 1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
<=
i
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) &
i
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
<=
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) holds
proj
(
i
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) ,
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) ) : ( (
Function-like
quasi_total
) ( non
empty
Relation-like
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-defined
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-valued
Function-like
total
quasi_total
V33
()
V34
()
V35
() )
Element
of
bool
[:
(
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) )) ,
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
:]
: ( ( ) ( )
set
) : ( ( ) ( )
set
) )
is_continuous_in
x0
: ( ( ) (
Relation-like
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) )
-defined
Function-like
V33
()
V34
()
V35
()
finite
V71
(
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) )
FinSequence-like
FinSubsequence-like
)
Element
of
REAL
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) )) ) ;
theorem
:: NFCONT_4:43
for
x0
being ( (
real
) (
V11
()
real
ext-real
)
number
)
for
n
being ( ( non
empty
) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
for
h
being ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
2
: ( ( non
empty
) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,) holds
(
h
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
2
: ( ( non
empty
) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,)
is_continuous_in
x0
: ( (
real
) (
V11
()
real
ext-real
)
number
) iff (
x0
: ( (
real
) (
V11
()
real
ext-real
)
number
)
in
dom
h
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
2
: ( ( non
empty
) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,) : ( ( ) ( )
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) & ( for
i
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) st
i
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
in
Seg
n
: ( ( non
empty
) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
finite
V71
(
b
2
: ( ( non
empty
) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) ) )
Element
of
bool
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) : ( ( ) ( )
set
) ) holds
(
proj
(
i
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) ,
n
: ( ( non
empty
) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) )
)
: ( (
Function-like
quasi_total
) ( non
empty
Relation-like
REAL
b
2
: ( ( non
empty
) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-defined
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-valued
Function-like
total
quasi_total
V33
()
V34
()
V35
() )
Element
of
bool
[:
(
REAL
b
2
: ( ( non
empty
) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) )) ,
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
:]
: ( ( ) ( )
set
) : ( ( ) ( )
set
) )
*
h
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
2
: ( ( non
empty
) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,) : ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-valued
Function-like
V33
()
V34
()
V35
() )
Element
of
bool
[:
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ,
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
:]
: ( ( ) ( )
set
) : ( ( ) ( )
set
) )
is_continuous_in
x0
: ( (
real
) (
V11
()
real
ext-real
)
number
) ) ) ) ;
theorem
:: NFCONT_4:44
for
n
being ( ( non
empty
) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
for
h
being ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( non
empty
) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,) holds
(
h
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( non
empty
) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,) is
continuous
iff for
i
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) st
i
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
in
Seg
n
: ( ( non
empty
) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
finite
V71
(
b
1
: ( ( non
empty
) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) ) )
Element
of
bool
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) : ( ( ) ( )
set
) ) holds
(
proj
(
i
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) ,
n
: ( ( non
empty
) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) )
)
: ( (
Function-like
quasi_total
) ( non
empty
Relation-like
REAL
b
1
: ( ( non
empty
) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-defined
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-valued
Function-like
total
quasi_total
V33
()
V34
()
V35
() )
Element
of
bool
[:
(
REAL
b
1
: ( ( non
empty
) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) )) ,
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
:]
: ( ( ) ( )
set
) : ( ( ) ( )
set
) )
*
h
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
b
1
: ( ( non
empty
) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
functional
FinSequence-membered
V227
()
V228
()
V229
() )
M12
(
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ))
-valued
Function-like
V233
()
V234
()
V235
() )
PartFunc
of ,) : ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-valued
Function-like
V33
()
V34
()
V35
() )
Element
of
bool
[:
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) ,
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
:]
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) is
continuous
) ;
theorem
:: NFCONT_4:45
for
n
,
i
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
for
x0
being ( ( ) ( )
Point
of ( ( ) ( non
empty
)
set
) ) st 1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
<=
i
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) &
i
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
<=
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) holds
Proj
(
i
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) ,
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) ) : ( (
Function-like
quasi_total
) ( non
empty
Relation-like
the
carrier
of
(
REAL-NS
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( non
empty
strict
) ( non
empty
V106
()
V131
()
V132
()
V133
()
V134
()
V135
()
V136
()
V137
()
V141
()
V142
()
strict
RealNormSpace-like
V174
() )
NORMSTR
) : ( ( ) ( non
empty
)
set
)
-defined
the
carrier
of
(
REAL-NS
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( non
empty
strict
) ( non
empty
non
trivial
V106
()
V131
()
V132
()
V133
()
V134
()
V135
()
V136
()
V137
()
V141
()
V142
()
strict
RealNormSpace-like
V174
() )
NORMSTR
) : ( ( ) ( non
empty
non
trivial
)
set
)
-valued
Function-like
total
quasi_total
)
Element
of
bool
[:
the
carrier
of
(
REAL-NS
b
1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( non
empty
strict
) ( non
empty
V106
()
V131
()
V132
()
V133
()
V134
()
V135
()
V136
()
V137
()
V141
()
V142
()
strict
RealNormSpace-like
V174
() )
NORMSTR
) : ( ( ) ( non
empty
)
set
) , the
carrier
of
(
REAL-NS
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( non
empty
strict
) ( non
empty
non
trivial
V106
()
V131
()
V132
()
V133
()
V134
()
V135
()
V136
()
V137
()
V141
()
V142
()
strict
RealNormSpace-like
V174
() )
NORMSTR
) : ( ( ) ( non
empty
non
trivial
)
set
)
:]
: ( ( ) ( )
set
) : ( ( ) ( )
set
) )
is_continuous_in
x0
: ( ( ) ( )
Point
of ( ( ) ( non
empty
)
set
) ) ;
theorem
:: NFCONT_4:46
for
x0
being ( (
real
) (
V11
()
real
ext-real
)
number
)
for
n
being ( ( non
empty
) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
for
h
being ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
the
carrier
of
(
REAL-NS
b
2
: ( ( non
empty
) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( non
empty
strict
) ( non
empty
non
trivial
V106
()
V131
()
V132
()
V133
()
V134
()
V135
()
V136
()
V137
()
V141
()
V142
()
strict
RealNormSpace-like
V174
() )
NORMSTR
) : ( ( ) ( non
empty
non
trivial
)
set
)
-valued
Function-like
)
PartFunc
of ,) holds
(
h
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
the
carrier
of
(
REAL-NS
b
2
: ( ( non
empty
) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( non
empty
strict
) ( non
empty
non
trivial
V106
()
V131
()
V132
()
V133
()
V134
()
V135
()
V136
()
V137
()
V141
()
V142
()
strict
RealNormSpace-like
V174
() )
NORMSTR
) : ( ( ) ( non
empty
non
trivial
)
set
)
-valued
Function-like
)
PartFunc
of ,)
is_continuous_in
x0
: ( (
real
) (
V11
()
real
ext-real
)
number
) iff for
i
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) st
i
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
in
Seg
n
: ( ( non
empty
) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
finite
V71
(
b
2
: ( ( non
empty
) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) ) )
Element
of
bool
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) : ( ( ) ( )
set
) ) holds
(
Proj
(
i
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) ,
n
: ( ( non
empty
) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) )
)
: ( (
Function-like
quasi_total
) ( non
empty
Relation-like
the
carrier
of
(
REAL-NS
b
2
: ( ( non
empty
) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( non
empty
strict
) ( non
empty
non
trivial
V106
()
V131
()
V132
()
V133
()
V134
()
V135
()
V136
()
V137
()
V141
()
V142
()
strict
RealNormSpace-like
V174
() )
NORMSTR
) : ( ( ) ( non
empty
non
trivial
)
set
)
-defined
the
carrier
of
(
REAL-NS
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( non
empty
strict
) ( non
empty
non
trivial
V106
()
V131
()
V132
()
V133
()
V134
()
V135
()
V136
()
V137
()
V141
()
V142
()
strict
RealNormSpace-like
V174
() )
NORMSTR
) : ( ( ) ( non
empty
non
trivial
)
set
)
-valued
Function-like
total
quasi_total
)
Element
of
bool
[:
the
carrier
of
(
REAL-NS
b
2
: ( ( non
empty
) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( non
empty
strict
) ( non
empty
non
trivial
V106
()
V131
()
V132
()
V133
()
V134
()
V135
()
V136
()
V137
()
V141
()
V142
()
strict
RealNormSpace-like
V174
() )
NORMSTR
) : ( ( ) ( non
empty
non
trivial
)
set
) , the
carrier
of
(
REAL-NS
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( non
empty
strict
) ( non
empty
non
trivial
V106
()
V131
()
V132
()
V133
()
V134
()
V135
()
V136
()
V137
()
V141
()
V142
()
strict
RealNormSpace-like
V174
() )
NORMSTR
) : ( ( ) ( non
empty
non
trivial
)
set
)
:]
: ( ( ) ( )
set
) : ( ( ) ( )
set
) )
*
h
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
the
carrier
of
(
REAL-NS
b
2
: ( ( non
empty
) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( non
empty
strict
) ( non
empty
non
trivial
V106
()
V131
()
V132
()
V133
()
V134
()
V135
()
V136
()
V137
()
V141
()
V142
()
strict
RealNormSpace-like
V174
() )
NORMSTR
) : ( ( ) ( non
empty
non
trivial
)
set
)
-valued
Function-like
)
PartFunc
of ,) : ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
the
carrier
of
(
REAL-NS
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( non
empty
strict
) ( non
empty
non
trivial
V106
()
V131
()
V132
()
V133
()
V134
()
V135
()
V136
()
V137
()
V141
()
V142
()
strict
RealNormSpace-like
V174
() )
NORMSTR
) : ( ( ) ( non
empty
non
trivial
)
set
)
-valued
Function-like
)
Element
of
bool
[:
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) , the
carrier
of
(
REAL-NS
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( non
empty
strict
) ( non
empty
non
trivial
V106
()
V131
()
V132
()
V133
()
V134
()
V135
()
V136
()
V137
()
V141
()
V142
()
strict
RealNormSpace-like
V174
() )
NORMSTR
) : ( ( ) ( non
empty
non
trivial
)
set
)
:]
: ( ( ) ( )
set
) : ( ( ) ( )
set
) )
is_continuous_in
x0
: ( (
real
) (
V11
()
real
ext-real
)
number
) ) ;
theorem
:: NFCONT_4:47
for
n
being ( ( non
empty
) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
for
h
being ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
the
carrier
of
(
REAL-NS
b
1
: ( ( non
empty
) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( non
empty
strict
) ( non
empty
non
trivial
V106
()
V131
()
V132
()
V133
()
V134
()
V135
()
V136
()
V137
()
V141
()
V142
()
strict
RealNormSpace-like
V174
() )
NORMSTR
) : ( ( ) ( non
empty
)
set
)
-valued
Function-like
)
PartFunc
of ,) holds
(
h
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
the
carrier
of
(
REAL-NS
b
1
: ( ( non
empty
) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( non
empty
strict
) ( non
empty
non
trivial
V106
()
V131
()
V132
()
V133
()
V134
()
V135
()
V136
()
V137
()
V141
()
V142
()
strict
RealNormSpace-like
V174
() )
NORMSTR
) : ( ( ) ( non
empty
)
set
)
-valued
Function-like
)
PartFunc
of ,) is
continuous
iff for
i
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) st
i
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
in
Seg
n
: ( ( non
empty
) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
finite
V71
(
b
1
: ( ( non
empty
) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) ) )
Element
of
bool
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) : ( ( ) ( )
set
) ) holds
(
Proj
(
i
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) ,
n
: ( ( non
empty
) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) ) )
)
: ( (
Function-like
quasi_total
) ( non
empty
Relation-like
the
carrier
of
(
REAL-NS
b
1
: ( ( non
empty
) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( non
empty
strict
) ( non
empty
non
trivial
V106
()
V131
()
V132
()
V133
()
V134
()
V135
()
V136
()
V137
()
V141
()
V142
()
strict
RealNormSpace-like
V174
() )
NORMSTR
) : ( ( ) ( non
empty
)
set
)
-defined
the
carrier
of
(
REAL-NS
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( non
empty
strict
) ( non
empty
non
trivial
V106
()
V131
()
V132
()
V133
()
V134
()
V135
()
V136
()
V137
()
V141
()
V142
()
strict
RealNormSpace-like
V174
() )
NORMSTR
) : ( ( ) ( non
empty
non
trivial
)
set
)
-valued
Function-like
total
quasi_total
)
Element
of
bool
[:
the
carrier
of
(
REAL-NS
b
1
: ( ( non
empty
) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( non
empty
strict
) ( non
empty
non
trivial
V106
()
V131
()
V132
()
V133
()
V134
()
V135
()
V136
()
V137
()
V141
()
V142
()
strict
RealNormSpace-like
V174
() )
NORMSTR
) : ( ( ) ( non
empty
)
set
) , the
carrier
of
(
REAL-NS
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( non
empty
strict
) ( non
empty
non
trivial
V106
()
V131
()
V132
()
V133
()
V134
()
V135
()
V136
()
V137
()
V141
()
V142
()
strict
RealNormSpace-like
V174
() )
NORMSTR
) : ( ( ) ( non
empty
non
trivial
)
set
)
:]
: ( ( ) ( )
set
) : ( ( ) ( )
set
) )
*
h
: ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
the
carrier
of
(
REAL-NS
b
1
: ( ( non
empty
) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( non
empty
strict
) ( non
empty
non
trivial
V106
()
V131
()
V132
()
V133
()
V134
()
V135
()
V136
()
V137
()
V141
()
V142
()
strict
RealNormSpace-like
V174
() )
NORMSTR
) : ( ( ) ( non
empty
)
set
)
-valued
Function-like
)
PartFunc
of ,) : ( (
Function-like
) (
Relation-like
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
)
-defined
the
carrier
of
(
REAL-NS
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( non
empty
strict
) ( non
empty
non
trivial
V106
()
V131
()
V132
()
V133
()
V134
()
V135
()
V136
()
V137
()
V141
()
V142
()
strict
RealNormSpace-like
V174
() )
NORMSTR
) : ( ( ) ( non
empty
non
trivial
)
set
)
-valued
Function-like
)
Element
of
bool
[:
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) , the
carrier
of
(
REAL-NS
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
)
Element
of
bool
REAL
: ( ( ) ( non
empty
non
finite
non
bounded_below
non
bounded_above
V68
() )
set
) : ( ( ) ( )
set
) ) )
)
: ( ( non
empty
strict
) ( non
empty
non
trivial
V106
()
V131
()
V132
()
V133
()
V134
()
V135
()
V136
()
V137
()
V141
()
V142
()
strict
RealNormSpace-like
V174
() )
NORMSTR
) : ( ( ) ( non
empty
non
trivial
)
set
)
:]
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) is
continuous
) ;