begin
definition
let f be ( (
V21() ) (
V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued )
PartFunc of ,) ;
let z be ( ( ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like )
Element of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ) ;
pred f is_hpartial_differentiable`11_in z means
ex
x0,
y0 being ( ( ) (
V11()
real ext-real )
Real) st
(
z : ( ( ) ( )
set )
= <*x0 : ( ( ) ( V11() real ext-real ) Real) ,y0 : ( ( ) ( V11() real ext-real ) Real) *> : ( ( ) ( )
set ) & ex
N being ( ( ) (
V160()
V161()
V162() )
Neighbourhood of
x0 : ( ( ) (
V11()
real ext-real )
Real) ) st
(
N : ( ( ) (
V160()
V161()
V162() )
Neighbourhood of
b2 : ( ( ) (
V11()
real ext-real )
Real) )
c= dom (SVF1 (1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ,(pdiff1 (f : ( ( ) ( ) set ) ,1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( V21() quasi_total ) ( non empty V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) , REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) ,z : ( ( ) ( ) set ) )) : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ,
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V160()
V161()
V162() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) & ex
L being ( (
V21()
linear ) (
V16()
V19(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued linear )
LinearFunc) ex
R being ( (
V21()
RestFunc-like ) (
V16()
V19(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued RestFunc-like )
RestFunc) st
for
x being ( ( ) (
V11()
real ext-real )
Real) st
x : ( ( ) (
V11()
real ext-real )
Real)
in N : ( ( ) (
V160()
V161()
V162() )
Neighbourhood of
b2 : ( ( ) (
V11()
real ext-real )
Real) ) holds
((SVF1 (1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ,(pdiff1 (f : ( ( ) ( ) set ) ,1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( V21() quasi_total ) ( non empty V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) , REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) ,z : ( ( ) ( ) set ) )) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued ) Element of K6(K7(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ,REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
- ((SVF1 (1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ,(pdiff1 (f : ( ( ) ( ) set ) ,1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( V21() quasi_total ) ( non empty V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) , REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) ,z : ( ( ) ( ) set ) )) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued ) Element of K6(K7(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ,REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) . x0 : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
= (L : ( ( V21() linear ) ( V16() V19( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued linear ) LinearFunc) . (x : ( ( ) ( V11() real ext-real ) Real) - x0 : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
+ (R : ( ( V21() RestFunc-like ) ( non empty V16() V19( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() total quasi_total complex-valued ext-real-valued real-valued RestFunc-like ) RestFunc) . (x : ( ( ) ( V11() real ext-real ) Real) - x0 : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) ) );
pred f is_hpartial_differentiable`12_in z means
ex
x0,
y0 being ( ( ) (
V11()
real ext-real )
Real) st
(
z : ( ( ) ( )
set )
= <*x0 : ( ( ) ( V11() real ext-real ) Real) ,y0 : ( ( ) ( V11() real ext-real ) Real) *> : ( ( ) ( )
set ) & ex
N being ( ( ) (
V160()
V161()
V162() )
Neighbourhood of
y0 : ( ( ) (
V11()
real ext-real )
Real) ) st
(
N : ( ( ) (
V160()
V161()
V162() )
Neighbourhood of
b2 : ( ( ) (
V11()
real ext-real )
Real) )
c= dom (SVF1 (2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ,(pdiff1 (f : ( ( ) ( ) set ) ,1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( V21() quasi_total ) ( non empty V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) , REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) ,z : ( ( ) ( ) set ) )) : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ,
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V160()
V161()
V162() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) & ex
L being ( (
V21()
linear ) (
V16()
V19(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued linear )
LinearFunc) ex
R being ( (
V21()
RestFunc-like ) (
V16()
V19(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued RestFunc-like )
RestFunc) st
for
y being ( ( ) (
V11()
real ext-real )
Real) st
y : ( ( ) (
V11()
real ext-real )
Real)
in N : ( ( ) (
V160()
V161()
V162() )
Neighbourhood of
b2 : ( ( ) (
V11()
real ext-real )
Real) ) holds
((SVF1 (2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ,(pdiff1 (f : ( ( ) ( ) set ) ,1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( V21() quasi_total ) ( non empty V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) , REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) ,z : ( ( ) ( ) set ) )) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued ) Element of K6(K7(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ,REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) . y : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
- ((SVF1 (2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ,(pdiff1 (f : ( ( ) ( ) set ) ,1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( V21() quasi_total ) ( non empty V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) , REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) ,z : ( ( ) ( ) set ) )) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued ) Element of K6(K7(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ,REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) . y0 : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
= (L : ( ( V21() linear ) ( V16() V19( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued linear ) LinearFunc) . (y : ( ( ) ( V11() real ext-real ) Real) - y0 : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
+ (R : ( ( V21() RestFunc-like ) ( non empty V16() V19( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() total quasi_total complex-valued ext-real-valued real-valued RestFunc-like ) RestFunc) . (y : ( ( ) ( V11() real ext-real ) Real) - y0 : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) ) );
pred f is_hpartial_differentiable`21_in z means
ex
x0,
y0 being ( ( ) (
V11()
real ext-real )
Real) st
(
z : ( ( ) ( )
set )
= <*x0 : ( ( ) ( V11() real ext-real ) Real) ,y0 : ( ( ) ( V11() real ext-real ) Real) *> : ( ( ) ( )
set ) & ex
N being ( ( ) (
V160()
V161()
V162() )
Neighbourhood of
x0 : ( ( ) (
V11()
real ext-real )
Real) ) st
(
N : ( ( ) (
V160()
V161()
V162() )
Neighbourhood of
b2 : ( ( ) (
V11()
real ext-real )
Real) )
c= dom (SVF1 (1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ,(pdiff1 (f : ( ( ) ( ) set ) ,2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( V21() quasi_total ) ( non empty V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) , REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) ,z : ( ( ) ( ) set ) )) : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ,
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V160()
V161()
V162() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) & ex
L being ( (
V21()
linear ) (
V16()
V19(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued linear )
LinearFunc) ex
R being ( (
V21()
RestFunc-like ) (
V16()
V19(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued RestFunc-like )
RestFunc) st
for
x being ( ( ) (
V11()
real ext-real )
Real) st
x : ( ( ) (
V11()
real ext-real )
Real)
in N : ( ( ) (
V160()
V161()
V162() )
Neighbourhood of
b2 : ( ( ) (
V11()
real ext-real )
Real) ) holds
((SVF1 (1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ,(pdiff1 (f : ( ( ) ( ) set ) ,2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( V21() quasi_total ) ( non empty V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) , REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) ,z : ( ( ) ( ) set ) )) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued ) Element of K6(K7(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ,REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
- ((SVF1 (1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ,(pdiff1 (f : ( ( ) ( ) set ) ,2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( V21() quasi_total ) ( non empty V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) , REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) ,z : ( ( ) ( ) set ) )) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued ) Element of K6(K7(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ,REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) . x0 : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
= (L : ( ( V21() linear ) ( V16() V19( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued linear ) LinearFunc) . (x : ( ( ) ( V11() real ext-real ) Real) - x0 : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
+ (R : ( ( V21() RestFunc-like ) ( non empty V16() V19( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() total quasi_total complex-valued ext-real-valued real-valued RestFunc-like ) RestFunc) . (x : ( ( ) ( V11() real ext-real ) Real) - x0 : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) ) );
pred f is_hpartial_differentiable`22_in z means
ex
x0,
y0 being ( ( ) (
V11()
real ext-real )
Real) st
(
z : ( ( ) ( )
set )
= <*x0 : ( ( ) ( V11() real ext-real ) Real) ,y0 : ( ( ) ( V11() real ext-real ) Real) *> : ( ( ) ( )
set ) & ex
N being ( ( ) (
V160()
V161()
V162() )
Neighbourhood of
y0 : ( ( ) (
V11()
real ext-real )
Real) ) st
(
N : ( ( ) (
V160()
V161()
V162() )
Neighbourhood of
b2 : ( ( ) (
V11()
real ext-real )
Real) )
c= dom (SVF1 (2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ,(pdiff1 (f : ( ( ) ( ) set ) ,2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( V21() quasi_total ) ( non empty V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) , REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) ,z : ( ( ) ( ) set ) )) : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ,
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V160()
V161()
V162() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) & ex
L being ( (
V21()
linear ) (
V16()
V19(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued linear )
LinearFunc) ex
R being ( (
V21()
RestFunc-like ) (
V16()
V19(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued RestFunc-like )
RestFunc) st
for
y being ( ( ) (
V11()
real ext-real )
Real) st
y : ( ( ) (
V11()
real ext-real )
Real)
in N : ( ( ) (
V160()
V161()
V162() )
Neighbourhood of
b2 : ( ( ) (
V11()
real ext-real )
Real) ) holds
((SVF1 (2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ,(pdiff1 (f : ( ( ) ( ) set ) ,2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( V21() quasi_total ) ( non empty V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) , REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) ,z : ( ( ) ( ) set ) )) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued ) Element of K6(K7(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ,REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) . y : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
- ((SVF1 (2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ,(pdiff1 (f : ( ( ) ( ) set ) ,2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( V21() quasi_total ) ( non empty V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) , REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) ,z : ( ( ) ( ) set ) )) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued ) Element of K6(K7(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ,REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) . y0 : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
= (L : ( ( V21() linear ) ( V16() V19( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued linear ) LinearFunc) . (y : ( ( ) ( V11() real ext-real ) Real) - y0 : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
+ (R : ( ( V21() RestFunc-like ) ( non empty V16() V19( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() total quasi_total complex-valued ext-real-valued real-valued RestFunc-like ) RestFunc) . (y : ( ( ) ( V11() real ext-real ) Real) - y0 : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) ) );
end;
definition
let f be ( (
V21() ) (
V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued )
PartFunc of ,) ;
let z be ( ( ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like )
Element of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ) ;
assume
f : ( (
V21() ) (
V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
is_hpartial_differentiable`11_in z : ( ( ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like )
Element of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
;
func hpartdiff11 (
f,
z)
-> ( ( ) (
V11()
real ext-real )
Real)
means
ex
x0,
y0 being ( ( ) (
V11()
real ext-real )
Real) st
(
z : ( ( ) ( )
set )
= <*x0 : ( ( ) ( V11() real ext-real ) Real) ,y0 : ( ( ) ( V11() real ext-real ) Real) *> : ( ( ) ( )
set ) & ex
N being ( ( ) (
V160()
V161()
V162() )
Neighbourhood of
x0 : ( ( ) (
V11()
real ext-real )
Real) ) st
(
N : ( ( ) (
V160()
V161()
V162() )
Neighbourhood of
b1 : ( ( ) (
V11()
real ext-real )
Real) )
c= dom (SVF1 (1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ,(pdiff1 (f : ( ( ) ( ) set ) ,1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( V21() quasi_total ) ( non empty V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) , REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) ,z : ( ( ) ( ) set ) )) : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ,
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V160()
V161()
V162() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) & ex
L being ( (
V21()
linear ) (
V16()
V19(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued linear )
LinearFunc) ex
R being ( (
V21()
RestFunc-like ) (
V16()
V19(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued RestFunc-like )
RestFunc) st
(
it : ( (
V21() ) (
V16()
V19(
REAL f : ( ( ) ( )
set ) : ( ( ) ( )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued )
Element of
K6(
K7(
(REAL f : ( ( ) ( ) set ) ) : ( ( ) ( )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ,
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
= L : ( (
V21()
linear ) (
V16()
V19(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued linear )
LinearFunc)
. 1 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) & ( for
x being ( ( ) (
V11()
real ext-real )
Real) st
x : ( ( ) (
V11()
real ext-real )
Real)
in N : ( ( ) (
V160()
V161()
V162() )
Neighbourhood of
b1 : ( ( ) (
V11()
real ext-real )
Real) ) holds
((SVF1 (1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ,(pdiff1 (f : ( ( ) ( ) set ) ,1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( V21() quasi_total ) ( non empty V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) , REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) ,z : ( ( ) ( ) set ) )) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued ) Element of K6(K7(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ,REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
- ((SVF1 (1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ,(pdiff1 (f : ( ( ) ( ) set ) ,1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( V21() quasi_total ) ( non empty V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) , REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) ,z : ( ( ) ( ) set ) )) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued ) Element of K6(K7(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ,REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) . x0 : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
= (L : ( ( V21() linear ) ( V16() V19( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued linear ) LinearFunc) . (x : ( ( ) ( V11() real ext-real ) Real) - x0 : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
+ (R : ( ( V21() RestFunc-like ) ( non empty V16() V19( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() total quasi_total complex-valued ext-real-valued real-valued RestFunc-like ) RestFunc) . (x : ( ( ) ( V11() real ext-real ) Real) - x0 : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) ) ) ) );
end;
definition
let f be ( (
V21() ) (
V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued )
PartFunc of ,) ;
let z be ( ( ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like )
Element of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ) ;
assume
f : ( (
V21() ) (
V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
is_hpartial_differentiable`12_in z : ( ( ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like )
Element of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
;
func hpartdiff12 (
f,
z)
-> ( ( ) (
V11()
real ext-real )
Real)
means
ex
x0,
y0 being ( ( ) (
V11()
real ext-real )
Real) st
(
z : ( ( ) ( )
set )
= <*x0 : ( ( ) ( V11() real ext-real ) Real) ,y0 : ( ( ) ( V11() real ext-real ) Real) *> : ( ( ) ( )
set ) & ex
N being ( ( ) (
V160()
V161()
V162() )
Neighbourhood of
y0 : ( ( ) (
V11()
real ext-real )
Real) ) st
(
N : ( ( ) (
V160()
V161()
V162() )
Neighbourhood of
b2 : ( ( ) (
V11()
real ext-real )
Real) )
c= dom (SVF1 (2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ,(pdiff1 (f : ( ( ) ( ) set ) ,1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( V21() quasi_total ) ( non empty V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) , REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) ,z : ( ( ) ( ) set ) )) : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ,
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V160()
V161()
V162() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) & ex
L being ( (
V21()
linear ) (
V16()
V19(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued linear )
LinearFunc) ex
R being ( (
V21()
RestFunc-like ) (
V16()
V19(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued RestFunc-like )
RestFunc) st
(
it : ( (
V21() ) (
V16()
V19(
REAL f : ( ( ) ( )
set ) : ( ( ) ( )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued )
Element of
K6(
K7(
(REAL f : ( ( ) ( ) set ) ) : ( ( ) ( )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ,
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
= L : ( (
V21()
linear ) (
V16()
V19(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued linear )
LinearFunc)
. 1 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) & ( for
y being ( ( ) (
V11()
real ext-real )
Real) st
y : ( ( ) (
V11()
real ext-real )
Real)
in N : ( ( ) (
V160()
V161()
V162() )
Neighbourhood of
b2 : ( ( ) (
V11()
real ext-real )
Real) ) holds
((SVF1 (2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ,(pdiff1 (f : ( ( ) ( ) set ) ,1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( V21() quasi_total ) ( non empty V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) , REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) ,z : ( ( ) ( ) set ) )) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued ) Element of K6(K7(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ,REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) . y : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
- ((SVF1 (2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ,(pdiff1 (f : ( ( ) ( ) set ) ,1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( V21() quasi_total ) ( non empty V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) , REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) ,z : ( ( ) ( ) set ) )) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued ) Element of K6(K7(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ,REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) . y0 : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
= (L : ( ( V21() linear ) ( V16() V19( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued linear ) LinearFunc) . (y : ( ( ) ( V11() real ext-real ) Real) - y0 : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
+ (R : ( ( V21() RestFunc-like ) ( non empty V16() V19( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() total quasi_total complex-valued ext-real-valued real-valued RestFunc-like ) RestFunc) . (y : ( ( ) ( V11() real ext-real ) Real) - y0 : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) ) ) ) );
end;
definition
let f be ( (
V21() ) (
V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued )
PartFunc of ,) ;
let z be ( ( ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like )
Element of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ) ;
assume
f : ( (
V21() ) (
V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
is_hpartial_differentiable`21_in z : ( ( ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like )
Element of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
;
func hpartdiff21 (
f,
z)
-> ( ( ) (
V11()
real ext-real )
Real)
means
ex
x0,
y0 being ( ( ) (
V11()
real ext-real )
Real) st
(
z : ( ( ) ( )
set )
= <*x0 : ( ( ) ( V11() real ext-real ) Real) ,y0 : ( ( ) ( V11() real ext-real ) Real) *> : ( ( ) ( )
set ) & ex
N being ( ( ) (
V160()
V161()
V162() )
Neighbourhood of
x0 : ( ( ) (
V11()
real ext-real )
Real) ) st
(
N : ( ( ) (
V160()
V161()
V162() )
Neighbourhood of
b1 : ( ( ) (
V11()
real ext-real )
Real) )
c= dom (SVF1 (1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ,(pdiff1 (f : ( ( ) ( ) set ) ,2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( V21() quasi_total ) ( non empty V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) , REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) ,z : ( ( ) ( ) set ) )) : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ,
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V160()
V161()
V162() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) & ex
L being ( (
V21()
linear ) (
V16()
V19(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued linear )
LinearFunc) ex
R being ( (
V21()
RestFunc-like ) (
V16()
V19(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued RestFunc-like )
RestFunc) st
(
it : ( (
V21() ) (
V16()
V19(
REAL f : ( ( ) ( )
set ) : ( ( ) ( )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued )
Element of
K6(
K7(
(REAL f : ( ( ) ( ) set ) ) : ( ( ) ( )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ,
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
= L : ( (
V21()
linear ) (
V16()
V19(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued linear )
LinearFunc)
. 1 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) & ( for
x being ( ( ) (
V11()
real ext-real )
Real) st
x : ( ( ) (
V11()
real ext-real )
Real)
in N : ( ( ) (
V160()
V161()
V162() )
Neighbourhood of
b1 : ( ( ) (
V11()
real ext-real )
Real) ) holds
((SVF1 (1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ,(pdiff1 (f : ( ( ) ( ) set ) ,2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( V21() quasi_total ) ( non empty V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) , REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) ,z : ( ( ) ( ) set ) )) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued ) Element of K6(K7(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ,REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
- ((SVF1 (1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ,(pdiff1 (f : ( ( ) ( ) set ) ,2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( V21() quasi_total ) ( non empty V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) , REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) ,z : ( ( ) ( ) set ) )) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued ) Element of K6(K7(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ,REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) . x0 : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
= (L : ( ( V21() linear ) ( V16() V19( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued linear ) LinearFunc) . (x : ( ( ) ( V11() real ext-real ) Real) - x0 : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
+ (R : ( ( V21() RestFunc-like ) ( non empty V16() V19( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() total quasi_total complex-valued ext-real-valued real-valued RestFunc-like ) RestFunc) . (x : ( ( ) ( V11() real ext-real ) Real) - x0 : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) ) ) ) );
end;
definition
let f be ( (
V21() ) (
V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued )
PartFunc of ,) ;
let z be ( ( ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like )
Element of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ) ;
assume
f : ( (
V21() ) (
V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
is_hpartial_differentiable`22_in z : ( ( ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like )
Element of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
;
func hpartdiff22 (
f,
z)
-> ( ( ) (
V11()
real ext-real )
Real)
means
ex
x0,
y0 being ( ( ) (
V11()
real ext-real )
Real) st
(
z : ( ( ) ( )
set )
= <*x0 : ( ( ) ( V11() real ext-real ) Real) ,y0 : ( ( ) ( V11() real ext-real ) Real) *> : ( ( ) ( )
set ) & ex
N being ( ( ) (
V160()
V161()
V162() )
Neighbourhood of
y0 : ( ( ) (
V11()
real ext-real )
Real) ) st
(
N : ( ( ) (
V160()
V161()
V162() )
Neighbourhood of
b2 : ( ( ) (
V11()
real ext-real )
Real) )
c= dom (SVF1 (2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ,(pdiff1 (f : ( ( ) ( ) set ) ,2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( V21() quasi_total ) ( non empty V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) , REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) ,z : ( ( ) ( ) set ) )) : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ,
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V160()
V161()
V162() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) & ex
L being ( (
V21()
linear ) (
V16()
V19(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued linear )
LinearFunc) ex
R being ( (
V21()
RestFunc-like ) (
V16()
V19(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued RestFunc-like )
RestFunc) st
(
it : ( (
V21() ) (
V16()
V19(
REAL f : ( ( ) ( )
set ) : ( ( ) ( )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued )
Element of
K6(
K7(
(REAL f : ( ( ) ( ) set ) ) : ( ( ) ( )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ,
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
= L : ( (
V21()
linear ) (
V16()
V19(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued linear )
LinearFunc)
. 1 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) & ( for
y being ( ( ) (
V11()
real ext-real )
Real) st
y : ( ( ) (
V11()
real ext-real )
Real)
in N : ( ( ) (
V160()
V161()
V162() )
Neighbourhood of
b2 : ( ( ) (
V11()
real ext-real )
Real) ) holds
((SVF1 (2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ,(pdiff1 (f : ( ( ) ( ) set ) ,2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( V21() quasi_total ) ( non empty V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) , REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) ,z : ( ( ) ( ) set ) )) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued ) Element of K6(K7(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ,REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) . y : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
- ((SVF1 (2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ,(pdiff1 (f : ( ( ) ( ) set ) ,2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( V21() quasi_total ) ( non empty V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) , REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) ,z : ( ( ) ( ) set ) )) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued ) Element of K6(K7(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ,REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) . y0 : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
= (L : ( ( V21() linear ) ( V16() V19( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued linear ) LinearFunc) . (y : ( ( ) ( V11() real ext-real ) Real) - y0 : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
+ (R : ( ( V21() RestFunc-like ) ( non empty V16() V19( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() total quasi_total complex-valued ext-real-valued real-valued RestFunc-like ) RestFunc) . (y : ( ( ) ( V11() real ext-real ) Real) - y0 : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) ) ) ) );
end;
theorem
for
x0,
y0 being ( ( ) (
V11()
real ext-real )
Real)
for
z being ( ( ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like )
Element of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
for
f being ( (
V21() ) (
V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued )
PartFunc of ,) st
z : ( ( ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like )
Element of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
= <*x0 : ( ( ) ( V11() real ext-real ) Real) ,y0 : ( ( ) ( V11() real ext-real ) Real) *> : ( ( ) ( )
set ) &
f : ( (
V21() ) (
V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
is_hpartial_differentiable`11_in z : ( ( ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like )
Element of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ) holds
SVF1 (1 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) ,
(pdiff1 (f : ( ( V21() ) ( V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( (
V21()
quasi_total ) ( non
empty V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
total quasi_total complex-valued ext-real-valued real-valued )
Function of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ,
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) ,
z : ( ( ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like )
Element of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ) ) : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ,
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_in x0 : ( ( ) (
V11()
real ext-real )
Real) ;
theorem
for
x0,
y0 being ( ( ) (
V11()
real ext-real )
Real)
for
z being ( ( ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like )
Element of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
for
f being ( (
V21() ) (
V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued )
PartFunc of ,) st
z : ( ( ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like )
Element of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
= <*x0 : ( ( ) ( V11() real ext-real ) Real) ,y0 : ( ( ) ( V11() real ext-real ) Real) *> : ( ( ) ( )
set ) &
f : ( (
V21() ) (
V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
is_hpartial_differentiable`12_in z : ( ( ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like )
Element of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ) holds
SVF1 (2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) ,
(pdiff1 (f : ( ( V21() ) ( V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( (
V21()
quasi_total ) ( non
empty V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
total quasi_total complex-valued ext-real-valued real-valued )
Function of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ,
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) ,
z : ( ( ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like )
Element of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ) ) : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ,
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_in y0 : ( ( ) (
V11()
real ext-real )
Real) ;
theorem
for
x0,
y0 being ( ( ) (
V11()
real ext-real )
Real)
for
z being ( ( ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like )
Element of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
for
f being ( (
V21() ) (
V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued )
PartFunc of ,) st
z : ( ( ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like )
Element of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
= <*x0 : ( ( ) ( V11() real ext-real ) Real) ,y0 : ( ( ) ( V11() real ext-real ) Real) *> : ( ( ) ( )
set ) &
f : ( (
V21() ) (
V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
is_hpartial_differentiable`21_in z : ( ( ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like )
Element of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ) holds
SVF1 (1 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) ,
(pdiff1 (f : ( ( V21() ) ( V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( (
V21()
quasi_total ) ( non
empty V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
total quasi_total complex-valued ext-real-valued real-valued )
Function of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ,
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) ,
z : ( ( ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like )
Element of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ) ) : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ,
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_in x0 : ( ( ) (
V11()
real ext-real )
Real) ;
theorem
for
x0,
y0 being ( ( ) (
V11()
real ext-real )
Real)
for
z being ( ( ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like )
Element of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
for
f being ( (
V21() ) (
V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued )
PartFunc of ,) st
z : ( ( ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like )
Element of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
= <*x0 : ( ( ) ( V11() real ext-real ) Real) ,y0 : ( ( ) ( V11() real ext-real ) Real) *> : ( ( ) ( )
set ) &
f : ( (
V21() ) (
V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
is_hpartial_differentiable`22_in z : ( ( ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like )
Element of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ) holds
SVF1 (2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) ,
(pdiff1 (f : ( ( V21() ) ( V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( (
V21()
quasi_total ) ( non
empty V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
total quasi_total complex-valued ext-real-valued real-valued )
Function of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ,
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) ,
z : ( ( ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like )
Element of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ) ) : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ,
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_in y0 : ( ( ) (
V11()
real ext-real )
Real) ;
theorem
for
x0,
y0 being ( ( ) (
V11()
real ext-real )
Real)
for
z being ( ( ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like )
Element of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
for
f being ( (
V21() ) (
V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued )
PartFunc of ,) st
z : ( ( ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like )
Element of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
= <*x0 : ( ( ) ( V11() real ext-real ) Real) ,y0 : ( ( ) ( V11() real ext-real ) Real) *> : ( ( ) ( )
set ) &
f : ( (
V21() ) (
V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
is_hpartial_differentiable`11_in z : ( ( ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like )
Element of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ) holds
hpartdiff11 (
f : ( (
V21() ) (
V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued )
PartFunc of ,) ,
z : ( ( ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like )
Element of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ) ) : ( ( ) (
V11()
real ext-real )
Real)
= diff (
(SVF1 (1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ,(pdiff1 (f : ( ( V21() ) ( V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( V21() quasi_total ) ( non empty V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) , REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) ,z : ( ( ) ( V16() V19( NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like ) Element of REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) )) : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ,
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) ,
x0 : ( ( ) (
V11()
real ext-real )
Real) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) ;
theorem
for
x0,
y0 being ( ( ) (
V11()
real ext-real )
Real)
for
z being ( ( ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like )
Element of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
for
f being ( (
V21() ) (
V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued )
PartFunc of ,) st
z : ( ( ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like )
Element of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
= <*x0 : ( ( ) ( V11() real ext-real ) Real) ,y0 : ( ( ) ( V11() real ext-real ) Real) *> : ( ( ) ( )
set ) &
f : ( (
V21() ) (
V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
is_hpartial_differentiable`12_in z : ( ( ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like )
Element of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ) holds
hpartdiff12 (
f : ( (
V21() ) (
V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued )
PartFunc of ,) ,
z : ( ( ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like )
Element of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ) ) : ( ( ) (
V11()
real ext-real )
Real)
= diff (
(SVF1 (2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ,(pdiff1 (f : ( ( V21() ) ( V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( V21() quasi_total ) ( non empty V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) , REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) ,z : ( ( ) ( V16() V19( NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like ) Element of REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) )) : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ,
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) ,
y0 : ( ( ) (
V11()
real ext-real )
Real) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) ;
theorem
for
x0,
y0 being ( ( ) (
V11()
real ext-real )
Real)
for
z being ( ( ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like )
Element of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
for
f being ( (
V21() ) (
V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued )
PartFunc of ,) st
z : ( ( ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like )
Element of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
= <*x0 : ( ( ) ( V11() real ext-real ) Real) ,y0 : ( ( ) ( V11() real ext-real ) Real) *> : ( ( ) ( )
set ) &
f : ( (
V21() ) (
V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
is_hpartial_differentiable`21_in z : ( ( ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like )
Element of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ) holds
hpartdiff21 (
f : ( (
V21() ) (
V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued )
PartFunc of ,) ,
z : ( ( ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like )
Element of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ) ) : ( ( ) (
V11()
real ext-real )
Real)
= diff (
(SVF1 (1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ,(pdiff1 (f : ( ( V21() ) ( V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( V21() quasi_total ) ( non empty V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) , REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) ,z : ( ( ) ( V16() V19( NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like ) Element of REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) )) : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ,
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) ,
x0 : ( ( ) (
V11()
real ext-real )
Real) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) ;
theorem
for
x0,
y0 being ( ( ) (
V11()
real ext-real )
Real)
for
z being ( ( ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like )
Element of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
for
f being ( (
V21() ) (
V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued )
PartFunc of ,) st
z : ( ( ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like )
Element of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
= <*x0 : ( ( ) ( V11() real ext-real ) Real) ,y0 : ( ( ) ( V11() real ext-real ) Real) *> : ( ( ) ( )
set ) &
f : ( (
V21() ) (
V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
is_hpartial_differentiable`22_in z : ( ( ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like )
Element of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ) holds
hpartdiff22 (
f : ( (
V21() ) (
V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued )
PartFunc of ,) ,
z : ( ( ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like )
Element of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ) ) : ( ( ) (
V11()
real ext-real )
Real)
= diff (
(SVF1 (2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ,(pdiff1 (f : ( ( V21() ) ( V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( V21() quasi_total ) ( non empty V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) , REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) ,z : ( ( ) ( V16() V19( NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like ) Element of REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) )) : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ,
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) ,
y0 : ( ( ) (
V11()
real ext-real )
Real) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) ;
begin
theorem
for
f being ( (
V21() ) (
V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
for
z0 being ( ( ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like )
Element of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
for
N being ( ( ) (
V160()
V161()
V162() )
Neighbourhood of
(proj (1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ,2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( (
V21()
quasi_total ) ( non
empty V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
total quasi_total complex-valued ext-real-valued real-valued )
Element of
K6(
K7(
(REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ,
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
. z0 : ( ( ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like )
Element of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) ) st
f : ( (
V21() ) (
V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
is_hpartial_differentiable`11_in z0 : ( ( ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like )
Element of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ) &
N : ( ( ) (
V160()
V161()
V162() )
Neighbourhood of
(proj (1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ,2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( (
V21()
quasi_total ) ( non
empty V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
total quasi_total complex-valued ext-real-valued real-valued )
Element of
K6(
K7(
(REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ,
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
. b2 : ( ( ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like )
Element of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) )
c= dom (SVF1 (1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ,(pdiff1 (f : ( ( V21() ) ( V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( V21() quasi_total ) ( non empty V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) , REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) ,z0 : ( ( ) ( V16() V19( NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like ) Element of REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) )) : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ,
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V160()
V161()
V162() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) holds
for
h being ( (
non-zero V21()
quasi_total 0 : ( ( ) (
empty ordinal natural V11()
real ext-real non
positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
-convergent ) (
V16()
non-zero V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
total quasi_total complex-valued ext-real-valued real-valued 0 : ( ( ) (
empty ordinal natural V11()
real ext-real non
positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
-convergent convergent )
Real_Sequence)
for
c being ( (
V21()
constant quasi_total ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
constant total quasi_total complex-valued ext-real-valued real-valued )
Real_Sequence) st
rng c : ( (
V21()
constant quasi_total ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
constant total quasi_total complex-valued ext-real-valued real-valued )
Real_Sequence) : ( ( ) (
V160()
V161()
V162() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) )
= {((proj (1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ,2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( V21() quasi_total ) ( non empty V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() total quasi_total complex-valued ext-real-valued real-valued ) Element of K6(K7((REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ,REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) . z0 : ( ( ) ( V16() V19( NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like ) Element of REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) } : ( ( ) ( non
empty V160()
V161()
V162() )
set ) &
rng (h : ( ( non-zero V21() quasi_total 0 : ( ( ) ( empty ordinal natural V11() real ext-real non positive non negative V136() V159() V160() V161() V162() V163() V164() V165() V166() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) -convergent ) ( V16() non-zero V19( NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() total quasi_total complex-valued ext-real-valued real-valued 0 : ( ( ) ( empty ordinal natural V11() real ext-real non positive non negative V136() V159() V160() V161() V162() V163() V164() V165() V166() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) -convergent convergent ) Real_Sequence) + c : ( ( V21() constant quasi_total ) ( V16() V19( NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() constant total quasi_total complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( (
V21() ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
total quasi_total complex-valued ext-real-valued real-valued )
Element of
K6(
K7(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V160()
V161()
V162() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) )
c= N : ( ( ) (
V160()
V161()
V162() )
Neighbourhood of
(proj (1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ,2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( (
V21()
quasi_total ) ( non
empty V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
total quasi_total complex-valued ext-real-valued real-valued )
Element of
K6(
K7(
(REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ,
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
. b2 : ( ( ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like )
Element of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) ) holds
(
(h : ( ( non-zero V21() quasi_total 0 : ( ( ) ( empty ordinal natural V11() real ext-real non positive non negative V136() V159() V160() V161() V162() V163() V164() V165() V166() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) -convergent ) ( V16() non-zero V19( NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() total quasi_total complex-valued ext-real-valued real-valued 0 : ( ( ) ( empty ordinal natural V11() real ext-real non positive non negative V136() V159() V160() V161() V162() V163() V164() V165() V166() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) -convergent convergent ) Real_Sequence) ") : ( (
V21() ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
total quasi_total complex-valued ext-real-valued real-valued )
Element of
K6(
K7(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
(#) (((SVF1 (1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ,(pdiff1 (f : ( ( V21() ) ( V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( V21() quasi_total ) ( non empty V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) , REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) ,z0 : ( ( ) ( V16() V19( NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like ) Element of REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) )) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued ) Element of K6(K7(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ,REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) /* (h : ( ( non-zero V21() quasi_total 0 : ( ( ) ( empty ordinal natural V11() real ext-real non positive non negative V136() V159() V160() V161() V162() V163() V164() V165() V166() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) -convergent ) ( V16() non-zero V19( NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() total quasi_total complex-valued ext-real-valued real-valued 0 : ( ( ) ( empty ordinal natural V11() real ext-real non positive non negative V136() V159() V160() V161() V162() V163() V164() V165() V166() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) -convergent convergent ) Real_Sequence) + c : ( ( V21() constant quasi_total ) ( V16() V19( NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() constant total quasi_total complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( V21() ) ( V16() V19( NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() total quasi_total complex-valued ext-real-valued real-valued ) Element of K6(K7(NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V21() quasi_total ) ( V16() V19( NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() total quasi_total complex-valued ext-real-valued real-valued ) Element of K6(K7(NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) - ((SVF1 (1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ,(pdiff1 (f : ( ( V21() ) ( V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( V21() quasi_total ) ( non empty V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) , REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) ,z0 : ( ( ) ( V16() V19( NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like ) Element of REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) )) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued ) Element of K6(K7(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ,REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) /* c : ( ( V21() constant quasi_total ) ( V16() V19( NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() constant total quasi_total complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( V21() quasi_total ) ( V16() V19( NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() total quasi_total complex-valued ext-real-valued real-valued ) Element of K6(K7(NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V21() ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
total quasi_total complex-valued ext-real-valued real-valued )
Element of
K6(
K7(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) : ( (
V21() ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
total quasi_total complex-valued ext-real-valued real-valued )
Element of
K6(
K7(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) is
convergent &
hpartdiff11 (
f : ( (
V21() ) (
V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued )
PartFunc of ,) ,
z0 : ( ( ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like )
Element of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ) ) : ( ( ) (
V11()
real ext-real )
Real)
= lim ((h : ( ( non-zero V21() quasi_total 0 : ( ( ) ( empty ordinal natural V11() real ext-real non positive non negative V136() V159() V160() V161() V162() V163() V164() V165() V166() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) -convergent ) ( V16() non-zero V19( NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() total quasi_total complex-valued ext-real-valued real-valued 0 : ( ( ) ( empty ordinal natural V11() real ext-real non positive non negative V136() V159() V160() V161() V162() V163() V164() V165() V166() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) -convergent convergent ) Real_Sequence) ") : ( ( V21() ) ( V16() V19( NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() total quasi_total complex-valued ext-real-valued real-valued ) Element of K6(K7(NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) (#) (((SVF1 (1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ,(pdiff1 (f : ( ( V21() ) ( V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( V21() quasi_total ) ( non empty V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) , REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) ,z0 : ( ( ) ( V16() V19( NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like ) Element of REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) )) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued ) Element of K6(K7(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ,REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) /* (h : ( ( non-zero V21() quasi_total 0 : ( ( ) ( empty ordinal natural V11() real ext-real non positive non negative V136() V159() V160() V161() V162() V163() V164() V165() V166() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) -convergent ) ( V16() non-zero V19( NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() total quasi_total complex-valued ext-real-valued real-valued 0 : ( ( ) ( empty ordinal natural V11() real ext-real non positive non negative V136() V159() V160() V161() V162() V163() V164() V165() V166() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) -convergent convergent ) Real_Sequence) + c : ( ( V21() constant quasi_total ) ( V16() V19( NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() constant total quasi_total complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( V21() ) ( V16() V19( NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() total quasi_total complex-valued ext-real-valued real-valued ) Element of K6(K7(NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V21() quasi_total ) ( V16() V19( NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() total quasi_total complex-valued ext-real-valued real-valued ) Element of K6(K7(NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) - ((SVF1 (1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ,(pdiff1 (f : ( ( V21() ) ( V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( V21() quasi_total ) ( non empty V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) , REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) ,z0 : ( ( ) ( V16() V19( NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like ) Element of REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) )) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued ) Element of K6(K7(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ,REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) /* c : ( ( V21() constant quasi_total ) ( V16() V19( NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() constant total quasi_total complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( V21() quasi_total ) ( V16() V19( NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() total quasi_total complex-valued ext-real-valued real-valued ) Element of K6(K7(NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V21() ) ( V16() V19( NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() total quasi_total complex-valued ext-real-valued real-valued ) Element of K6(K7(NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V21() ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
total quasi_total complex-valued ext-real-valued real-valued )
Element of
K6(
K7(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) ) ;
theorem
for
f being ( (
V21() ) (
V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
for
z0 being ( ( ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like )
Element of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
for
N being ( ( ) (
V160()
V161()
V162() )
Neighbourhood of
(proj (2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ,2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( (
V21()
quasi_total ) ( non
empty V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
total quasi_total complex-valued ext-real-valued real-valued )
Element of
K6(
K7(
(REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ,
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
. z0 : ( ( ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like )
Element of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) ) st
f : ( (
V21() ) (
V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
is_hpartial_differentiable`12_in z0 : ( ( ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like )
Element of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ) &
N : ( ( ) (
V160()
V161()
V162() )
Neighbourhood of
(proj (2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ,2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( (
V21()
quasi_total ) ( non
empty V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
total quasi_total complex-valued ext-real-valued real-valued )
Element of
K6(
K7(
(REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ,
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
. b2 : ( ( ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like )
Element of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) )
c= dom (SVF1 (2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ,(pdiff1 (f : ( ( V21() ) ( V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( V21() quasi_total ) ( non empty V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) , REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) ,z0 : ( ( ) ( V16() V19( NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like ) Element of REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) )) : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ,
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V160()
V161()
V162() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) holds
for
h being ( (
non-zero V21()
quasi_total 0 : ( ( ) (
empty ordinal natural V11()
real ext-real non
positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
-convergent ) (
V16()
non-zero V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
total quasi_total complex-valued ext-real-valued real-valued 0 : ( ( ) (
empty ordinal natural V11()
real ext-real non
positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
-convergent convergent )
Real_Sequence)
for
c being ( (
V21()
constant quasi_total ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
constant total quasi_total complex-valued ext-real-valued real-valued )
Real_Sequence) st
rng c : ( (
V21()
constant quasi_total ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
constant total quasi_total complex-valued ext-real-valued real-valued )
Real_Sequence) : ( ( ) (
V160()
V161()
V162() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) )
= {((proj (2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ,2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( V21() quasi_total ) ( non empty V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() total quasi_total complex-valued ext-real-valued real-valued ) Element of K6(K7((REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ,REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) . z0 : ( ( ) ( V16() V19( NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like ) Element of REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) } : ( ( ) ( non
empty V160()
V161()
V162() )
set ) &
rng (h : ( ( non-zero V21() quasi_total 0 : ( ( ) ( empty ordinal natural V11() real ext-real non positive non negative V136() V159() V160() V161() V162() V163() V164() V165() V166() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) -convergent ) ( V16() non-zero V19( NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() total quasi_total complex-valued ext-real-valued real-valued 0 : ( ( ) ( empty ordinal natural V11() real ext-real non positive non negative V136() V159() V160() V161() V162() V163() V164() V165() V166() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) -convergent convergent ) Real_Sequence) + c : ( ( V21() constant quasi_total ) ( V16() V19( NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() constant total quasi_total complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( (
V21() ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
total quasi_total complex-valued ext-real-valued real-valued )
Element of
K6(
K7(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V160()
V161()
V162() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) )
c= N : ( ( ) (
V160()
V161()
V162() )
Neighbourhood of
(proj (2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ,2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( (
V21()
quasi_total ) ( non
empty V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
total quasi_total complex-valued ext-real-valued real-valued )
Element of
K6(
K7(
(REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ,
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
. b2 : ( ( ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like )
Element of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) ) holds
(
(h : ( ( non-zero V21() quasi_total 0 : ( ( ) ( empty ordinal natural V11() real ext-real non positive non negative V136() V159() V160() V161() V162() V163() V164() V165() V166() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) -convergent ) ( V16() non-zero V19( NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() total quasi_total complex-valued ext-real-valued real-valued 0 : ( ( ) ( empty ordinal natural V11() real ext-real non positive non negative V136() V159() V160() V161() V162() V163() V164() V165() V166() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) -convergent convergent ) Real_Sequence) ") : ( (
V21() ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
total quasi_total complex-valued ext-real-valued real-valued )
Element of
K6(
K7(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
(#) (((SVF1 (2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ,(pdiff1 (f : ( ( V21() ) ( V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( V21() quasi_total ) ( non empty V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) , REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) ,z0 : ( ( ) ( V16() V19( NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like ) Element of REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) )) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued ) Element of K6(K7(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ,REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) /* (h : ( ( non-zero V21() quasi_total 0 : ( ( ) ( empty ordinal natural V11() real ext-real non positive non negative V136() V159() V160() V161() V162() V163() V164() V165() V166() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) -convergent ) ( V16() non-zero V19( NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() total quasi_total complex-valued ext-real-valued real-valued 0 : ( ( ) ( empty ordinal natural V11() real ext-real non positive non negative V136() V159() V160() V161() V162() V163() V164() V165() V166() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) -convergent convergent ) Real_Sequence) + c : ( ( V21() constant quasi_total ) ( V16() V19( NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() constant total quasi_total complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( V21() ) ( V16() V19( NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() total quasi_total complex-valued ext-real-valued real-valued ) Element of K6(K7(NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V21() quasi_total ) ( V16() V19( NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() total quasi_total complex-valued ext-real-valued real-valued ) Element of K6(K7(NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) - ((SVF1 (2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ,(pdiff1 (f : ( ( V21() ) ( V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( V21() quasi_total ) ( non empty V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) , REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) ,z0 : ( ( ) ( V16() V19( NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like ) Element of REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) )) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued ) Element of K6(K7(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ,REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) /* c : ( ( V21() constant quasi_total ) ( V16() V19( NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() constant total quasi_total complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( V21() quasi_total ) ( V16() V19( NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() total quasi_total complex-valued ext-real-valued real-valued ) Element of K6(K7(NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V21() ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
total quasi_total complex-valued ext-real-valued real-valued )
Element of
K6(
K7(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) : ( (
V21() ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
total quasi_total complex-valued ext-real-valued real-valued )
Element of
K6(
K7(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) is
convergent &
hpartdiff12 (
f : ( (
V21() ) (
V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued )
PartFunc of ,) ,
z0 : ( ( ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like )
Element of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ) ) : ( ( ) (
V11()
real ext-real )
Real)
= lim ((h : ( ( non-zero V21() quasi_total 0 : ( ( ) ( empty ordinal natural V11() real ext-real non positive non negative V136() V159() V160() V161() V162() V163() V164() V165() V166() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) -convergent ) ( V16() non-zero V19( NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() total quasi_total complex-valued ext-real-valued real-valued 0 : ( ( ) ( empty ordinal natural V11() real ext-real non positive non negative V136() V159() V160() V161() V162() V163() V164() V165() V166() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) -convergent convergent ) Real_Sequence) ") : ( ( V21() ) ( V16() V19( NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() total quasi_total complex-valued ext-real-valued real-valued ) Element of K6(K7(NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) (#) (((SVF1 (2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ,(pdiff1 (f : ( ( V21() ) ( V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( V21() quasi_total ) ( non empty V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) , REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) ,z0 : ( ( ) ( V16() V19( NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like ) Element of REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) )) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued ) Element of K6(K7(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ,REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) /* (h : ( ( non-zero V21() quasi_total 0 : ( ( ) ( empty ordinal natural V11() real ext-real non positive non negative V136() V159() V160() V161() V162() V163() V164() V165() V166() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) -convergent ) ( V16() non-zero V19( NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() total quasi_total complex-valued ext-real-valued real-valued 0 : ( ( ) ( empty ordinal natural V11() real ext-real non positive non negative V136() V159() V160() V161() V162() V163() V164() V165() V166() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) -convergent convergent ) Real_Sequence) + c : ( ( V21() constant quasi_total ) ( V16() V19( NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() constant total quasi_total complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( V21() ) ( V16() V19( NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() total quasi_total complex-valued ext-real-valued real-valued ) Element of K6(K7(NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V21() quasi_total ) ( V16() V19( NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() total quasi_total complex-valued ext-real-valued real-valued ) Element of K6(K7(NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) - ((SVF1 (2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ,(pdiff1 (f : ( ( V21() ) ( V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( V21() quasi_total ) ( non empty V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) , REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) ,z0 : ( ( ) ( V16() V19( NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like ) Element of REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) )) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued ) Element of K6(K7(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ,REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) /* c : ( ( V21() constant quasi_total ) ( V16() V19( NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() constant total quasi_total complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( V21() quasi_total ) ( V16() V19( NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() total quasi_total complex-valued ext-real-valued real-valued ) Element of K6(K7(NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V21() ) ( V16() V19( NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() total quasi_total complex-valued ext-real-valued real-valued ) Element of K6(K7(NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V21() ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
total quasi_total complex-valued ext-real-valued real-valued )
Element of
K6(
K7(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) ) ;
theorem
for
f being ( (
V21() ) (
V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
for
z0 being ( ( ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like )
Element of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
for
N being ( ( ) (
V160()
V161()
V162() )
Neighbourhood of
(proj (1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ,2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( (
V21()
quasi_total ) ( non
empty V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
total quasi_total complex-valued ext-real-valued real-valued )
Element of
K6(
K7(
(REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ,
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
. z0 : ( ( ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like )
Element of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) ) st
f : ( (
V21() ) (
V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
is_hpartial_differentiable`21_in z0 : ( ( ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like )
Element of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ) &
N : ( ( ) (
V160()
V161()
V162() )
Neighbourhood of
(proj (1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ,2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( (
V21()
quasi_total ) ( non
empty V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
total quasi_total complex-valued ext-real-valued real-valued )
Element of
K6(
K7(
(REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ,
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
. b2 : ( ( ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like )
Element of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) )
c= dom (SVF1 (1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ,(pdiff1 (f : ( ( V21() ) ( V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( V21() quasi_total ) ( non empty V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) , REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) ,z0 : ( ( ) ( V16() V19( NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like ) Element of REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) )) : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ,
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V160()
V161()
V162() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) holds
for
h being ( (
non-zero V21()
quasi_total 0 : ( ( ) (
empty ordinal natural V11()
real ext-real non
positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
-convergent ) (
V16()
non-zero V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
total quasi_total complex-valued ext-real-valued real-valued 0 : ( ( ) (
empty ordinal natural V11()
real ext-real non
positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
-convergent convergent )
Real_Sequence)
for
c being ( (
V21()
constant quasi_total ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
constant total quasi_total complex-valued ext-real-valued real-valued )
Real_Sequence) st
rng c : ( (
V21()
constant quasi_total ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
constant total quasi_total complex-valued ext-real-valued real-valued )
Real_Sequence) : ( ( ) (
V160()
V161()
V162() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) )
= {((proj (1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ,2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( V21() quasi_total ) ( non empty V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() total quasi_total complex-valued ext-real-valued real-valued ) Element of K6(K7((REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ,REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) . z0 : ( ( ) ( V16() V19( NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like ) Element of REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) } : ( ( ) ( non
empty V160()
V161()
V162() )
set ) &
rng (h : ( ( non-zero V21() quasi_total 0 : ( ( ) ( empty ordinal natural V11() real ext-real non positive non negative V136() V159() V160() V161() V162() V163() V164() V165() V166() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) -convergent ) ( V16() non-zero V19( NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() total quasi_total complex-valued ext-real-valued real-valued 0 : ( ( ) ( empty ordinal natural V11() real ext-real non positive non negative V136() V159() V160() V161() V162() V163() V164() V165() V166() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) -convergent convergent ) Real_Sequence) + c : ( ( V21() constant quasi_total ) ( V16() V19( NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() constant total quasi_total complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( (
V21() ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
total quasi_total complex-valued ext-real-valued real-valued )
Element of
K6(
K7(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V160()
V161()
V162() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) )
c= N : ( ( ) (
V160()
V161()
V162() )
Neighbourhood of
(proj (1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ,2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( (
V21()
quasi_total ) ( non
empty V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
total quasi_total complex-valued ext-real-valued real-valued )
Element of
K6(
K7(
(REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ,
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
. b2 : ( ( ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like )
Element of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) ) holds
(
(h : ( ( non-zero V21() quasi_total 0 : ( ( ) ( empty ordinal natural V11() real ext-real non positive non negative V136() V159() V160() V161() V162() V163() V164() V165() V166() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) -convergent ) ( V16() non-zero V19( NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() total quasi_total complex-valued ext-real-valued real-valued 0 : ( ( ) ( empty ordinal natural V11() real ext-real non positive non negative V136() V159() V160() V161() V162() V163() V164() V165() V166() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) -convergent convergent ) Real_Sequence) ") : ( (
V21() ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
total quasi_total complex-valued ext-real-valued real-valued )
Element of
K6(
K7(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
(#) (((SVF1 (1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ,(pdiff1 (f : ( ( V21() ) ( V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( V21() quasi_total ) ( non empty V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) , REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) ,z0 : ( ( ) ( V16() V19( NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like ) Element of REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) )) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued ) Element of K6(K7(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ,REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) /* (h : ( ( non-zero V21() quasi_total 0 : ( ( ) ( empty ordinal natural V11() real ext-real non positive non negative V136() V159() V160() V161() V162() V163() V164() V165() V166() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) -convergent ) ( V16() non-zero V19( NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() total quasi_total complex-valued ext-real-valued real-valued 0 : ( ( ) ( empty ordinal natural V11() real ext-real non positive non negative V136() V159() V160() V161() V162() V163() V164() V165() V166() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) -convergent convergent ) Real_Sequence) + c : ( ( V21() constant quasi_total ) ( V16() V19( NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() constant total quasi_total complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( V21() ) ( V16() V19( NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() total quasi_total complex-valued ext-real-valued real-valued ) Element of K6(K7(NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V21() quasi_total ) ( V16() V19( NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() total quasi_total complex-valued ext-real-valued real-valued ) Element of K6(K7(NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) - ((SVF1 (1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ,(pdiff1 (f : ( ( V21() ) ( V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( V21() quasi_total ) ( non empty V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) , REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) ,z0 : ( ( ) ( V16() V19( NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like ) Element of REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) )) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued ) Element of K6(K7(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ,REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) /* c : ( ( V21() constant quasi_total ) ( V16() V19( NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() constant total quasi_total complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( V21() quasi_total ) ( V16() V19( NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() total quasi_total complex-valued ext-real-valued real-valued ) Element of K6(K7(NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V21() ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
total quasi_total complex-valued ext-real-valued real-valued )
Element of
K6(
K7(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) : ( (
V21() ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
total quasi_total complex-valued ext-real-valued real-valued )
Element of
K6(
K7(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) is
convergent &
hpartdiff21 (
f : ( (
V21() ) (
V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued )
PartFunc of ,) ,
z0 : ( ( ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like )
Element of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ) ) : ( ( ) (
V11()
real ext-real )
Real)
= lim ((h : ( ( non-zero V21() quasi_total 0 : ( ( ) ( empty ordinal natural V11() real ext-real non positive non negative V136() V159() V160() V161() V162() V163() V164() V165() V166() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) -convergent ) ( V16() non-zero V19( NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() total quasi_total complex-valued ext-real-valued real-valued 0 : ( ( ) ( empty ordinal natural V11() real ext-real non positive non negative V136() V159() V160() V161() V162() V163() V164() V165() V166() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) -convergent convergent ) Real_Sequence) ") : ( ( V21() ) ( V16() V19( NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() total quasi_total complex-valued ext-real-valued real-valued ) Element of K6(K7(NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) (#) (((SVF1 (1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ,(pdiff1 (f : ( ( V21() ) ( V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( V21() quasi_total ) ( non empty V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) , REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) ,z0 : ( ( ) ( V16() V19( NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like ) Element of REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) )) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued ) Element of K6(K7(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ,REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) /* (h : ( ( non-zero V21() quasi_total 0 : ( ( ) ( empty ordinal natural V11() real ext-real non positive non negative V136() V159() V160() V161() V162() V163() V164() V165() V166() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) -convergent ) ( V16() non-zero V19( NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() total quasi_total complex-valued ext-real-valued real-valued 0 : ( ( ) ( empty ordinal natural V11() real ext-real non positive non negative V136() V159() V160() V161() V162() V163() V164() V165() V166() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) -convergent convergent ) Real_Sequence) + c : ( ( V21() constant quasi_total ) ( V16() V19( NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() constant total quasi_total complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( V21() ) ( V16() V19( NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() total quasi_total complex-valued ext-real-valued real-valued ) Element of K6(K7(NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V21() quasi_total ) ( V16() V19( NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() total quasi_total complex-valued ext-real-valued real-valued ) Element of K6(K7(NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) - ((SVF1 (1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ,(pdiff1 (f : ( ( V21() ) ( V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( V21() quasi_total ) ( non empty V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) , REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) ,z0 : ( ( ) ( V16() V19( NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like ) Element of REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) )) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued ) Element of K6(K7(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ,REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) /* c : ( ( V21() constant quasi_total ) ( V16() V19( NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() constant total quasi_total complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( V21() quasi_total ) ( V16() V19( NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() total quasi_total complex-valued ext-real-valued real-valued ) Element of K6(K7(NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V21() ) ( V16() V19( NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() total quasi_total complex-valued ext-real-valued real-valued ) Element of K6(K7(NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V21() ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
total quasi_total complex-valued ext-real-valued real-valued )
Element of
K6(
K7(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) ) ;
theorem
for
f being ( (
V21() ) (
V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
for
z0 being ( ( ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like )
Element of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
for
N being ( ( ) (
V160()
V161()
V162() )
Neighbourhood of
(proj (2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ,2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( (
V21()
quasi_total ) ( non
empty V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
total quasi_total complex-valued ext-real-valued real-valued )
Element of
K6(
K7(
(REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ,
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
. z0 : ( ( ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like )
Element of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) ) st
f : ( (
V21() ) (
V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
is_hpartial_differentiable`22_in z0 : ( ( ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like )
Element of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ) &
N : ( ( ) (
V160()
V161()
V162() )
Neighbourhood of
(proj (2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ,2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( (
V21()
quasi_total ) ( non
empty V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
total quasi_total complex-valued ext-real-valued real-valued )
Element of
K6(
K7(
(REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ,
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
. b2 : ( ( ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like )
Element of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) )
c= dom (SVF1 (2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ,(pdiff1 (f : ( ( V21() ) ( V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( V21() quasi_total ) ( non empty V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) , REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) ,z0 : ( ( ) ( V16() V19( NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like ) Element of REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) )) : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ,
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V160()
V161()
V162() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) holds
for
h being ( (
non-zero V21()
quasi_total 0 : ( ( ) (
empty ordinal natural V11()
real ext-real non
positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
-convergent ) (
V16()
non-zero V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
total quasi_total complex-valued ext-real-valued real-valued 0 : ( ( ) (
empty ordinal natural V11()
real ext-real non
positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
-convergent convergent )
Real_Sequence)
for
c being ( (
V21()
constant quasi_total ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
constant total quasi_total complex-valued ext-real-valued real-valued )
Real_Sequence) st
rng c : ( (
V21()
constant quasi_total ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
constant total quasi_total complex-valued ext-real-valued real-valued )
Real_Sequence) : ( ( ) (
V160()
V161()
V162() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) )
= {((proj (2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ,2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( V21() quasi_total ) ( non empty V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() total quasi_total complex-valued ext-real-valued real-valued ) Element of K6(K7((REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ,REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) . z0 : ( ( ) ( V16() V19( NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like ) Element of REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) } : ( ( ) ( non
empty V160()
V161()
V162() )
set ) &
rng (h : ( ( non-zero V21() quasi_total 0 : ( ( ) ( empty ordinal natural V11() real ext-real non positive non negative V136() V159() V160() V161() V162() V163() V164() V165() V166() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) -convergent ) ( V16() non-zero V19( NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() total quasi_total complex-valued ext-real-valued real-valued 0 : ( ( ) ( empty ordinal natural V11() real ext-real non positive non negative V136() V159() V160() V161() V162() V163() V164() V165() V166() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) -convergent convergent ) Real_Sequence) + c : ( ( V21() constant quasi_total ) ( V16() V19( NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() constant total quasi_total complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( (
V21() ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
total quasi_total complex-valued ext-real-valued real-valued )
Element of
K6(
K7(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V160()
V161()
V162() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) )
c= N : ( ( ) (
V160()
V161()
V162() )
Neighbourhood of
(proj (2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ,2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( (
V21()
quasi_total ) ( non
empty V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
total quasi_total complex-valued ext-real-valued real-valued )
Element of
K6(
K7(
(REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ,
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
. b2 : ( ( ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like )
Element of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) ) holds
(
(h : ( ( non-zero V21() quasi_total 0 : ( ( ) ( empty ordinal natural V11() real ext-real non positive non negative V136() V159() V160() V161() V162() V163() V164() V165() V166() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) -convergent ) ( V16() non-zero V19( NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() total quasi_total complex-valued ext-real-valued real-valued 0 : ( ( ) ( empty ordinal natural V11() real ext-real non positive non negative V136() V159() V160() V161() V162() V163() V164() V165() V166() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) -convergent convergent ) Real_Sequence) ") : ( (
V21() ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
total quasi_total complex-valued ext-real-valued real-valued )
Element of
K6(
K7(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
(#) (((SVF1 (2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ,(pdiff1 (f : ( ( V21() ) ( V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( V21() quasi_total ) ( non empty V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) , REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) ,z0 : ( ( ) ( V16() V19( NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like ) Element of REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) )) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued ) Element of K6(K7(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ,REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) /* (h : ( ( non-zero V21() quasi_total 0 : ( ( ) ( empty ordinal natural V11() real ext-real non positive non negative V136() V159() V160() V161() V162() V163() V164() V165() V166() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) -convergent ) ( V16() non-zero V19( NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() total quasi_total complex-valued ext-real-valued real-valued 0 : ( ( ) ( empty ordinal natural V11() real ext-real non positive non negative V136() V159() V160() V161() V162() V163() V164() V165() V166() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) -convergent convergent ) Real_Sequence) + c : ( ( V21() constant quasi_total ) ( V16() V19( NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() constant total quasi_total complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( V21() ) ( V16() V19( NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() total quasi_total complex-valued ext-real-valued real-valued ) Element of K6(K7(NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V21() quasi_total ) ( V16() V19( NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() total quasi_total complex-valued ext-real-valued real-valued ) Element of K6(K7(NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) - ((SVF1 (2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ,(pdiff1 (f : ( ( V21() ) ( V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( V21() quasi_total ) ( non empty V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) , REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) ,z0 : ( ( ) ( V16() V19( NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like ) Element of REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) )) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued ) Element of K6(K7(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ,REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) /* c : ( ( V21() constant quasi_total ) ( V16() V19( NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() constant total quasi_total complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( V21() quasi_total ) ( V16() V19( NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() total quasi_total complex-valued ext-real-valued real-valued ) Element of K6(K7(NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V21() ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
total quasi_total complex-valued ext-real-valued real-valued )
Element of
K6(
K7(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) : ( (
V21() ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
total quasi_total complex-valued ext-real-valued real-valued )
Element of
K6(
K7(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) is
convergent &
hpartdiff22 (
f : ( (
V21() ) (
V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued )
PartFunc of ,) ,
z0 : ( ( ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like )
Element of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ) ) : ( ( ) (
V11()
real ext-real )
Real)
= lim ((h : ( ( non-zero V21() quasi_total 0 : ( ( ) ( empty ordinal natural V11() real ext-real non positive non negative V136() V159() V160() V161() V162() V163() V164() V165() V166() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) -convergent ) ( V16() non-zero V19( NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() total quasi_total complex-valued ext-real-valued real-valued 0 : ( ( ) ( empty ordinal natural V11() real ext-real non positive non negative V136() V159() V160() V161() V162() V163() V164() V165() V166() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) -convergent convergent ) Real_Sequence) ") : ( ( V21() ) ( V16() V19( NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() total quasi_total complex-valued ext-real-valued real-valued ) Element of K6(K7(NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) (#) (((SVF1 (2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ,(pdiff1 (f : ( ( V21() ) ( V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( V21() quasi_total ) ( non empty V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) , REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) ,z0 : ( ( ) ( V16() V19( NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like ) Element of REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) )) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued ) Element of K6(K7(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ,REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) /* (h : ( ( non-zero V21() quasi_total 0 : ( ( ) ( empty ordinal natural V11() real ext-real non positive non negative V136() V159() V160() V161() V162() V163() V164() V165() V166() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) -convergent ) ( V16() non-zero V19( NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() total quasi_total complex-valued ext-real-valued real-valued 0 : ( ( ) ( empty ordinal natural V11() real ext-real non positive non negative V136() V159() V160() V161() V162() V163() V164() V165() V166() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) -convergent convergent ) Real_Sequence) + c : ( ( V21() constant quasi_total ) ( V16() V19( NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() constant total quasi_total complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( V21() ) ( V16() V19( NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() total quasi_total complex-valued ext-real-valued real-valued ) Element of K6(K7(NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V21() quasi_total ) ( V16() V19( NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() total quasi_total complex-valued ext-real-valued real-valued ) Element of K6(K7(NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) - ((SVF1 (2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ,(pdiff1 (f : ( ( V21() ) ( V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( V21() quasi_total ) ( non empty V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) , REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) ,z0 : ( ( ) ( V16() V19( NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like ) Element of REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) )) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued ) Element of K6(K7(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ,REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) /* c : ( ( V21() constant quasi_total ) ( V16() V19( NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() constant total quasi_total complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( V21() quasi_total ) ( V16() V19( NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() total quasi_total complex-valued ext-real-valued real-valued ) Element of K6(K7(NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V21() ) ( V16() V19( NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() total quasi_total complex-valued ext-real-valued real-valued ) Element of K6(K7(NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V21() ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
total quasi_total complex-valued ext-real-valued real-valued )
Element of
K6(
K7(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) ) ;
theorem
for
z0 being ( ( ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like )
Element of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
for
f1,
f2 being ( (
V21() ) (
V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued )
PartFunc of ,) st
f1 : ( (
V21() ) (
V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
is_hpartial_differentiable`11_in z0 : ( ( ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like )
Element of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ) &
f2 : ( (
V21() ) (
V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
is_hpartial_differentiable`11_in z0 : ( ( ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like )
Element of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ) holds
(
(pdiff1 (f1 : ( ( V21() ) ( V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( (
V21()
quasi_total ) ( non
empty V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
total quasi_total complex-valued ext-real-valued real-valued )
Function of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ,
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
+ (pdiff1 (f2 : ( ( V21() ) ( V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( (
V21()
quasi_total ) ( non
empty V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
total quasi_total complex-valued ext-real-valued real-valued )
Function of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ,
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( (
V21() ) ( non
empty V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
total quasi_total complex-valued ext-real-valued real-valued )
Element of
K6(
K7(
(REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ,
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
is_partial_differentiable_in z0 : ( ( ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like )
Element of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ) ,1 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) &
partdiff (
((pdiff1 (f1 : ( ( V21() ) ( V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( V21() quasi_total ) ( non empty V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) , REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) + (pdiff1 (f2 : ( ( V21() ) ( V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( V21() quasi_total ) ( non empty V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) , REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) ) : ( (
V21() ) ( non
empty V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
total quasi_total complex-valued ext-real-valued real-valued )
Element of
K6(
K7(
(REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ,
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) ,
z0 : ( ( ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like )
Element of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ) ,1 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
= (hpartdiff11 (f1 : ( ( V21() ) ( V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,z0 : ( ( ) ( V16() V19( NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like ) Element of REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) )) : ( ( ) (
V11()
real ext-real )
Real)
+ (hpartdiff11 (f2 : ( ( V21() ) ( V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,z0 : ( ( ) ( V16() V19( NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like ) Element of REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) )) : ( ( ) (
V11()
real ext-real )
Real) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) ) ;
theorem
for
z0 being ( ( ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like )
Element of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
for
f1,
f2 being ( (
V21() ) (
V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued )
PartFunc of ,) st
f1 : ( (
V21() ) (
V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
is_hpartial_differentiable`12_in z0 : ( ( ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like )
Element of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ) &
f2 : ( (
V21() ) (
V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
is_hpartial_differentiable`12_in z0 : ( ( ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like )
Element of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ) holds
(
(pdiff1 (f1 : ( ( V21() ) ( V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( (
V21()
quasi_total ) ( non
empty V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
total quasi_total complex-valued ext-real-valued real-valued )
Function of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ,
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
+ (pdiff1 (f2 : ( ( V21() ) ( V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( (
V21()
quasi_total ) ( non
empty V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
total quasi_total complex-valued ext-real-valued real-valued )
Function of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ,
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( (
V21() ) ( non
empty V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
total quasi_total complex-valued ext-real-valued real-valued )
Element of
K6(
K7(
(REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ,
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
is_partial_differentiable_in z0 : ( ( ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like )
Element of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ) ,2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) &
partdiff (
((pdiff1 (f1 : ( ( V21() ) ( V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( V21() quasi_total ) ( non empty V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) , REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) + (pdiff1 (f2 : ( ( V21() ) ( V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( V21() quasi_total ) ( non empty V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) , REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) ) : ( (
V21() ) ( non
empty V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
total quasi_total complex-valued ext-real-valued real-valued )
Element of
K6(
K7(
(REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ,
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) ,
z0 : ( ( ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like )
Element of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ) ,2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
= (hpartdiff12 (f1 : ( ( V21() ) ( V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,z0 : ( ( ) ( V16() V19( NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like ) Element of REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) )) : ( ( ) (
V11()
real ext-real )
Real)
+ (hpartdiff12 (f2 : ( ( V21() ) ( V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,z0 : ( ( ) ( V16() V19( NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like ) Element of REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) )) : ( ( ) (
V11()
real ext-real )
Real) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) ) ;
theorem
for
z0 being ( ( ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like )
Element of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
for
f1,
f2 being ( (
V21() ) (
V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued )
PartFunc of ,) st
f1 : ( (
V21() ) (
V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
is_hpartial_differentiable`21_in z0 : ( ( ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like )
Element of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ) &
f2 : ( (
V21() ) (
V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
is_hpartial_differentiable`21_in z0 : ( ( ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like )
Element of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ) holds
(
(pdiff1 (f1 : ( ( V21() ) ( V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( (
V21()
quasi_total ) ( non
empty V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
total quasi_total complex-valued ext-real-valued real-valued )
Function of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ,
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
+ (pdiff1 (f2 : ( ( V21() ) ( V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( (
V21()
quasi_total ) ( non
empty V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
total quasi_total complex-valued ext-real-valued real-valued )
Function of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ,
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( (
V21() ) ( non
empty V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
total quasi_total complex-valued ext-real-valued real-valued )
Element of
K6(
K7(
(REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ,
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
is_partial_differentiable_in z0 : ( ( ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like )
Element of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ) ,1 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) &
partdiff (
((pdiff1 (f1 : ( ( V21() ) ( V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( V21() quasi_total ) ( non empty V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) , REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) + (pdiff1 (f2 : ( ( V21() ) ( V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( V21() quasi_total ) ( non empty V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) , REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) ) : ( (
V21() ) ( non
empty V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
total quasi_total complex-valued ext-real-valued real-valued )
Element of
K6(
K7(
(REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ,
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) ,
z0 : ( ( ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like )
Element of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ) ,1 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
= (hpartdiff21 (f1 : ( ( V21() ) ( V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,z0 : ( ( ) ( V16() V19( NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like ) Element of REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) )) : ( ( ) (
V11()
real ext-real )
Real)
+ (hpartdiff21 (f2 : ( ( V21() ) ( V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,z0 : ( ( ) ( V16() V19( NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like ) Element of REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) )) : ( ( ) (
V11()
real ext-real )
Real) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) ) ;
theorem
for
z0 being ( ( ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like )
Element of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
for
f1,
f2 being ( (
V21() ) (
V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued )
PartFunc of ,) st
f1 : ( (
V21() ) (
V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
is_hpartial_differentiable`22_in z0 : ( ( ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like )
Element of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ) &
f2 : ( (
V21() ) (
V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
is_hpartial_differentiable`22_in z0 : ( ( ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like )
Element of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ) holds
(
(pdiff1 (f1 : ( ( V21() ) ( V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( (
V21()
quasi_total ) ( non
empty V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
total quasi_total complex-valued ext-real-valued real-valued )
Function of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ,
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
+ (pdiff1 (f2 : ( ( V21() ) ( V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( (
V21()
quasi_total ) ( non
empty V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
total quasi_total complex-valued ext-real-valued real-valued )
Function of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ,
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( (
V21() ) ( non
empty V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
total quasi_total complex-valued ext-real-valued real-valued )
Element of
K6(
K7(
(REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ,
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
is_partial_differentiable_in z0 : ( ( ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like )
Element of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ) ,2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) &
partdiff (
((pdiff1 (f1 : ( ( V21() ) ( V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( V21() quasi_total ) ( non empty V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) , REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) + (pdiff1 (f2 : ( ( V21() ) ( V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( V21() quasi_total ) ( non empty V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) , REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) ) : ( (
V21() ) ( non
empty V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
total quasi_total complex-valued ext-real-valued real-valued )
Element of
K6(
K7(
(REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ,
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) ,
z0 : ( ( ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like )
Element of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ) ,2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
= (hpartdiff22 (f1 : ( ( V21() ) ( V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,z0 : ( ( ) ( V16() V19( NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like ) Element of REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) )) : ( ( ) (
V11()
real ext-real )
Real)
+ (hpartdiff22 (f2 : ( ( V21() ) ( V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,z0 : ( ( ) ( V16() V19( NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like ) Element of REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) )) : ( ( ) (
V11()
real ext-real )
Real) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) ) ;
theorem
for
z0 being ( ( ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like )
Element of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
for
f1,
f2 being ( (
V21() ) (
V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued )
PartFunc of ,) st
f1 : ( (
V21() ) (
V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
is_hpartial_differentiable`11_in z0 : ( ( ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like )
Element of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ) &
f2 : ( (
V21() ) (
V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
is_hpartial_differentiable`11_in z0 : ( ( ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like )
Element of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ) holds
(
(pdiff1 (f1 : ( ( V21() ) ( V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( (
V21()
quasi_total ) ( non
empty V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
total quasi_total complex-valued ext-real-valued real-valued )
Function of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ,
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
- (pdiff1 (f2 : ( ( V21() ) ( V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( (
V21()
quasi_total ) ( non
empty V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
total quasi_total complex-valued ext-real-valued real-valued )
Function of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ,
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( (
V21() ) ( non
empty V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
total quasi_total complex-valued ext-real-valued real-valued )
Element of
K6(
K7(
(REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ,
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
is_partial_differentiable_in z0 : ( ( ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like )
Element of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ) ,1 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) &
partdiff (
((pdiff1 (f1 : ( ( V21() ) ( V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( V21() quasi_total ) ( non empty V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) , REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) - (pdiff1 (f2 : ( ( V21() ) ( V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( V21() quasi_total ) ( non empty V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) , REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) ) : ( (
V21() ) ( non
empty V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
total quasi_total complex-valued ext-real-valued real-valued )
Element of
K6(
K7(
(REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ,
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) ,
z0 : ( ( ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like )
Element of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ) ,1 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
= (hpartdiff11 (f1 : ( ( V21() ) ( V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,z0 : ( ( ) ( V16() V19( NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like ) Element of REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) )) : ( ( ) (
V11()
real ext-real )
Real)
- (hpartdiff11 (f2 : ( ( V21() ) ( V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,z0 : ( ( ) ( V16() V19( NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like ) Element of REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) )) : ( ( ) (
V11()
real ext-real )
Real) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) ) ;
theorem
for
z0 being ( ( ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like )
Element of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
for
f1,
f2 being ( (
V21() ) (
V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued )
PartFunc of ,) st
f1 : ( (
V21() ) (
V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
is_hpartial_differentiable`12_in z0 : ( ( ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like )
Element of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ) &
f2 : ( (
V21() ) (
V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
is_hpartial_differentiable`12_in z0 : ( ( ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like )
Element of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ) holds
(
(pdiff1 (f1 : ( ( V21() ) ( V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( (
V21()
quasi_total ) ( non
empty V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
total quasi_total complex-valued ext-real-valued real-valued )
Function of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ,
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
- (pdiff1 (f2 : ( ( V21() ) ( V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( (
V21()
quasi_total ) ( non
empty V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
total quasi_total complex-valued ext-real-valued real-valued )
Function of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ,
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( (
V21() ) ( non
empty V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
total quasi_total complex-valued ext-real-valued real-valued )
Element of
K6(
K7(
(REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ,
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
is_partial_differentiable_in z0 : ( ( ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like )
Element of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ) ,2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) &
partdiff (
((pdiff1 (f1 : ( ( V21() ) ( V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( V21() quasi_total ) ( non empty V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) , REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) - (pdiff1 (f2 : ( ( V21() ) ( V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( V21() quasi_total ) ( non empty V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) , REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) ) : ( (
V21() ) ( non
empty V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
total quasi_total complex-valued ext-real-valued real-valued )
Element of
K6(
K7(
(REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ,
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) ,
z0 : ( ( ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like )
Element of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ) ,2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
= (hpartdiff12 (f1 : ( ( V21() ) ( V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,z0 : ( ( ) ( V16() V19( NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like ) Element of REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) )) : ( ( ) (
V11()
real ext-real )
Real)
- (hpartdiff12 (f2 : ( ( V21() ) ( V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,z0 : ( ( ) ( V16() V19( NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like ) Element of REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) )) : ( ( ) (
V11()
real ext-real )
Real) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) ) ;
theorem
for
z0 being ( ( ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like )
Element of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
for
f1,
f2 being ( (
V21() ) (
V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued )
PartFunc of ,) st
f1 : ( (
V21() ) (
V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
is_hpartial_differentiable`21_in z0 : ( ( ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like )
Element of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ) &
f2 : ( (
V21() ) (
V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
is_hpartial_differentiable`21_in z0 : ( ( ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like )
Element of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ) holds
(
(pdiff1 (f1 : ( ( V21() ) ( V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( (
V21()
quasi_total ) ( non
empty V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
total quasi_total complex-valued ext-real-valued real-valued )
Function of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ,
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
- (pdiff1 (f2 : ( ( V21() ) ( V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( (
V21()
quasi_total ) ( non
empty V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
total quasi_total complex-valued ext-real-valued real-valued )
Function of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ,
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( (
V21() ) ( non
empty V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
total quasi_total complex-valued ext-real-valued real-valued )
Element of
K6(
K7(
(REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ,
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
is_partial_differentiable_in z0 : ( ( ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like )
Element of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ) ,1 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) &
partdiff (
((pdiff1 (f1 : ( ( V21() ) ( V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( V21() quasi_total ) ( non empty V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) , REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) - (pdiff1 (f2 : ( ( V21() ) ( V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( V21() quasi_total ) ( non empty V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) , REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) ) : ( (
V21() ) ( non
empty V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
total quasi_total complex-valued ext-real-valued real-valued )
Element of
K6(
K7(
(REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ,
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) ,
z0 : ( ( ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like )
Element of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ) ,1 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
= (hpartdiff21 (f1 : ( ( V21() ) ( V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,z0 : ( ( ) ( V16() V19( NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like ) Element of REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) )) : ( ( ) (
V11()
real ext-real )
Real)
- (hpartdiff21 (f2 : ( ( V21() ) ( V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,z0 : ( ( ) ( V16() V19( NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like ) Element of REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) )) : ( ( ) (
V11()
real ext-real )
Real) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) ) ;
theorem
for
z0 being ( ( ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like )
Element of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
for
f1,
f2 being ( (
V21() ) (
V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued )
PartFunc of ,) st
f1 : ( (
V21() ) (
V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
is_hpartial_differentiable`22_in z0 : ( ( ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like )
Element of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ) &
f2 : ( (
V21() ) (
V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
is_hpartial_differentiable`22_in z0 : ( ( ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like )
Element of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ) holds
(
(pdiff1 (f1 : ( ( V21() ) ( V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( (
V21()
quasi_total ) ( non
empty V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
total quasi_total complex-valued ext-real-valued real-valued )
Function of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ,
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
- (pdiff1 (f2 : ( ( V21() ) ( V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( (
V21()
quasi_total ) ( non
empty V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
total quasi_total complex-valued ext-real-valued real-valued )
Function of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ,
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( (
V21() ) ( non
empty V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
total quasi_total complex-valued ext-real-valued real-valued )
Element of
K6(
K7(
(REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ,
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
is_partial_differentiable_in z0 : ( ( ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like )
Element of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ) ,2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) &
partdiff (
((pdiff1 (f1 : ( ( V21() ) ( V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( V21() quasi_total ) ( non empty V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) , REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) - (pdiff1 (f2 : ( ( V21() ) ( V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( V21() quasi_total ) ( non empty V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) , REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) ) : ( (
V21() ) ( non
empty V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
total quasi_total complex-valued ext-real-valued real-valued )
Element of
K6(
K7(
(REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ,
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) ,
z0 : ( ( ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like )
Element of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ) ,2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
= (hpartdiff22 (f1 : ( ( V21() ) ( V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,z0 : ( ( ) ( V16() V19( NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like ) Element of REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) )) : ( ( ) (
V11()
real ext-real )
Real)
- (hpartdiff22 (f2 : ( ( V21() ) ( V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,z0 : ( ( ) ( V16() V19( NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like ) Element of REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) )) : ( ( ) (
V11()
real ext-real )
Real) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) ) ;
theorem
for
r being ( ( ) (
V11()
real ext-real )
Real)
for
z0 being ( ( ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like )
Element of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
for
f being ( (
V21() ) (
V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued )
PartFunc of ,) st
f : ( (
V21() ) (
V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
is_hpartial_differentiable`11_in z0 : ( ( ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like )
Element of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ) holds
(
r : ( ( ) (
V11()
real ext-real )
Real)
(#) (pdiff1 (f : ( ( V21() ) ( V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( (
V21()
quasi_total ) ( non
empty V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
total quasi_total complex-valued ext-real-valued real-valued )
Function of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ,
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( (
V21() ) ( non
empty V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
total quasi_total complex-valued ext-real-valued real-valued )
Element of
K6(
K7(
(REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ,
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
is_partial_differentiable_in z0 : ( ( ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like )
Element of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ) ,1 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) &
partdiff (
(r : ( ( ) ( V11() real ext-real ) Real) (#) (pdiff1 (f : ( ( V21() ) ( V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( V21() quasi_total ) ( non empty V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) , REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) ) : ( (
V21() ) ( non
empty V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
total quasi_total complex-valued ext-real-valued real-valued )
Element of
K6(
K7(
(REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ,
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) ,
z0 : ( ( ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like )
Element of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ) ,1 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
= r : ( ( ) (
V11()
real ext-real )
Real)
* (hpartdiff11 (f : ( ( V21() ) ( V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,z0 : ( ( ) ( V16() V19( NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like ) Element of REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) )) : ( ( ) (
V11()
real ext-real )
Real) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) ) ;
theorem
for
r being ( ( ) (
V11()
real ext-real )
Real)
for
z0 being ( ( ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like )
Element of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
for
f being ( (
V21() ) (
V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued )
PartFunc of ,) st
f : ( (
V21() ) (
V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
is_hpartial_differentiable`12_in z0 : ( ( ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like )
Element of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ) holds
(
r : ( ( ) (
V11()
real ext-real )
Real)
(#) (pdiff1 (f : ( ( V21() ) ( V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( (
V21()
quasi_total ) ( non
empty V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
total quasi_total complex-valued ext-real-valued real-valued )
Function of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ,
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( (
V21() ) ( non
empty V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
total quasi_total complex-valued ext-real-valued real-valued )
Element of
K6(
K7(
(REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ,
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
is_partial_differentiable_in z0 : ( ( ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like )
Element of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ) ,2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) &
partdiff (
(r : ( ( ) ( V11() real ext-real ) Real) (#) (pdiff1 (f : ( ( V21() ) ( V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( V21() quasi_total ) ( non empty V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) , REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) ) : ( (
V21() ) ( non
empty V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
total quasi_total complex-valued ext-real-valued real-valued )
Element of
K6(
K7(
(REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ,
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) ,
z0 : ( ( ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like )
Element of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ) ,2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
= r : ( ( ) (
V11()
real ext-real )
Real)
* (hpartdiff12 (f : ( ( V21() ) ( V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,z0 : ( ( ) ( V16() V19( NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like ) Element of REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) )) : ( ( ) (
V11()
real ext-real )
Real) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) ) ;
theorem
for
r being ( ( ) (
V11()
real ext-real )
Real)
for
z0 being ( ( ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like )
Element of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
for
f being ( (
V21() ) (
V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued )
PartFunc of ,) st
f : ( (
V21() ) (
V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
is_hpartial_differentiable`21_in z0 : ( ( ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like )
Element of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ) holds
(
r : ( ( ) (
V11()
real ext-real )
Real)
(#) (pdiff1 (f : ( ( V21() ) ( V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( (
V21()
quasi_total ) ( non
empty V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
total quasi_total complex-valued ext-real-valued real-valued )
Function of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ,
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( (
V21() ) ( non
empty V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
total quasi_total complex-valued ext-real-valued real-valued )
Element of
K6(
K7(
(REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ,
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
is_partial_differentiable_in z0 : ( ( ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like )
Element of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ) ,1 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) &
partdiff (
(r : ( ( ) ( V11() real ext-real ) Real) (#) (pdiff1 (f : ( ( V21() ) ( V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( V21() quasi_total ) ( non empty V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) , REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) ) : ( (
V21() ) ( non
empty V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
total quasi_total complex-valued ext-real-valued real-valued )
Element of
K6(
K7(
(REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ,
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) ,
z0 : ( ( ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like )
Element of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ) ,1 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
= r : ( ( ) (
V11()
real ext-real )
Real)
* (hpartdiff21 (f : ( ( V21() ) ( V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,z0 : ( ( ) ( V16() V19( NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like ) Element of REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) )) : ( ( ) (
V11()
real ext-real )
Real) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) ) ;
theorem
for
r being ( ( ) (
V11()
real ext-real )
Real)
for
z0 being ( ( ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like )
Element of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
for
f being ( (
V21() ) (
V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued )
PartFunc of ,) st
f : ( (
V21() ) (
V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
is_hpartial_differentiable`22_in z0 : ( ( ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like )
Element of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ) holds
(
r : ( ( ) (
V11()
real ext-real )
Real)
(#) (pdiff1 (f : ( ( V21() ) ( V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( (
V21()
quasi_total ) ( non
empty V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
total quasi_total complex-valued ext-real-valued real-valued )
Function of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ,
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( (
V21() ) ( non
empty V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
total quasi_total complex-valued ext-real-valued real-valued )
Element of
K6(
K7(
(REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ,
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
is_partial_differentiable_in z0 : ( ( ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like )
Element of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ) ,2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) &
partdiff (
(r : ( ( ) ( V11() real ext-real ) Real) (#) (pdiff1 (f : ( ( V21() ) ( V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( V21() quasi_total ) ( non empty V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) , REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) ) : ( (
V21() ) ( non
empty V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
total quasi_total complex-valued ext-real-valued real-valued )
Element of
K6(
K7(
(REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ,
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) ,
z0 : ( ( ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like )
Element of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ) ,2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
= r : ( ( ) (
V11()
real ext-real )
Real)
* (hpartdiff22 (f : ( ( V21() ) ( V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,z0 : ( ( ) ( V16() V19( NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like ) Element of REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) )) : ( ( ) (
V11()
real ext-real )
Real) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) ) ;
theorem
for
z0 being ( ( ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like )
Element of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
for
f1,
f2 being ( (
V21() ) (
V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued )
PartFunc of ,) st
f1 : ( (
V21() ) (
V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
is_hpartial_differentiable`11_in z0 : ( ( ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like )
Element of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ) &
f2 : ( (
V21() ) (
V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
is_hpartial_differentiable`11_in z0 : ( ( ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like )
Element of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ) holds
(pdiff1 (f1 : ( ( V21() ) ( V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( (
V21()
quasi_total ) ( non
empty V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
total quasi_total complex-valued ext-real-valued real-valued )
Function of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ,
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
(#) (pdiff1 (f2 : ( ( V21() ) ( V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( (
V21()
quasi_total ) ( non
empty V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
total quasi_total complex-valued ext-real-valued real-valued )
Function of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ,
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( (
V21() ) ( non
empty V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
total quasi_total complex-valued ext-real-valued real-valued )
Element of
K6(
K7(
(REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ,
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
is_partial_differentiable_in z0 : ( ( ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like )
Element of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ) ,1 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) ;
theorem
for
z0 being ( ( ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like )
Element of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
for
f1,
f2 being ( (
V21() ) (
V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued )
PartFunc of ,) st
f1 : ( (
V21() ) (
V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
is_hpartial_differentiable`12_in z0 : ( ( ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like )
Element of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ) &
f2 : ( (
V21() ) (
V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
is_hpartial_differentiable`12_in z0 : ( ( ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like )
Element of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ) holds
(pdiff1 (f1 : ( ( V21() ) ( V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( (
V21()
quasi_total ) ( non
empty V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
total quasi_total complex-valued ext-real-valued real-valued )
Function of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ,
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
(#) (pdiff1 (f2 : ( ( V21() ) ( V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( (
V21()
quasi_total ) ( non
empty V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
total quasi_total complex-valued ext-real-valued real-valued )
Function of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ,
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( (
V21() ) ( non
empty V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
total quasi_total complex-valued ext-real-valued real-valued )
Element of
K6(
K7(
(REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ,
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
is_partial_differentiable_in z0 : ( ( ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like )
Element of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ) ,2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) ;
theorem
for
z0 being ( ( ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like )
Element of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
for
f1,
f2 being ( (
V21() ) (
V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued )
PartFunc of ,) st
f1 : ( (
V21() ) (
V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
is_hpartial_differentiable`21_in z0 : ( ( ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like )
Element of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ) &
f2 : ( (
V21() ) (
V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
is_hpartial_differentiable`21_in z0 : ( ( ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like )
Element of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ) holds
(pdiff1 (f1 : ( ( V21() ) ( V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( (
V21()
quasi_total ) ( non
empty V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
total quasi_total complex-valued ext-real-valued real-valued )
Function of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ,
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
(#) (pdiff1 (f2 : ( ( V21() ) ( V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( (
V21()
quasi_total ) ( non
empty V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
total quasi_total complex-valued ext-real-valued real-valued )
Function of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ,
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( (
V21() ) ( non
empty V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
total quasi_total complex-valued ext-real-valued real-valued )
Element of
K6(
K7(
(REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ,
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
is_partial_differentiable_in z0 : ( ( ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like )
Element of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ) ,1 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) ;
theorem
for
z0 being ( ( ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like )
Element of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
for
f1,
f2 being ( (
V21() ) (
V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued )
PartFunc of ,) st
f1 : ( (
V21() ) (
V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
is_hpartial_differentiable`22_in z0 : ( ( ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like )
Element of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ) &
f2 : ( (
V21() ) (
V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
is_hpartial_differentiable`22_in z0 : ( ( ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like )
Element of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ) holds
(pdiff1 (f1 : ( ( V21() ) ( V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( (
V21()
quasi_total ) ( non
empty V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
total quasi_total complex-valued ext-real-valued real-valued )
Function of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ,
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
(#) (pdiff1 (f2 : ( ( V21() ) ( V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( (
V21()
quasi_total ) ( non
empty V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
total quasi_total complex-valued ext-real-valued real-valued )
Function of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ,
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( (
V21() ) ( non
empty V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
total quasi_total complex-valued ext-real-valued real-valued )
Element of
K6(
K7(
(REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ,
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
is_partial_differentiable_in z0 : ( ( ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like )
Element of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ) ,2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) ;
theorem
for
f being ( (
V21() ) (
V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
for
z0 being ( ( ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like )
Element of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ) st
f : ( (
V21() ) (
V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
is_hpartial_differentiable`11_in z0 : ( ( ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like )
Element of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ) holds
SVF1 (1 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) ,
(pdiff1 (f : ( ( V21() ) ( V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( (
V21()
quasi_total ) ( non
empty V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
total quasi_total complex-valued ext-real-valued real-valued )
Function of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ,
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) ,
z0 : ( ( ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like )
Element of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ) ) : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ,
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
is_continuous_in (proj (1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ,2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( (
V21()
quasi_total ) ( non
empty V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
total quasi_total complex-valued ext-real-valued real-valued )
Element of
K6(
K7(
(REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ,
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
. z0 : ( ( ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like )
Element of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) ;
theorem
for
f being ( (
V21() ) (
V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
for
z0 being ( ( ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like )
Element of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ) st
f : ( (
V21() ) (
V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
is_hpartial_differentiable`12_in z0 : ( ( ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like )
Element of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ) holds
SVF1 (2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) ,
(pdiff1 (f : ( ( V21() ) ( V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( (
V21()
quasi_total ) ( non
empty V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
total quasi_total complex-valued ext-real-valued real-valued )
Function of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ,
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) ,
z0 : ( ( ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like )
Element of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ) ) : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ,
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
is_continuous_in (proj (2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ,2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( (
V21()
quasi_total ) ( non
empty V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
total quasi_total complex-valued ext-real-valued real-valued )
Element of
K6(
K7(
(REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ,
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
. z0 : ( ( ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like )
Element of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) ;
theorem
for
f being ( (
V21() ) (
V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
for
z0 being ( ( ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like )
Element of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ) st
f : ( (
V21() ) (
V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
is_hpartial_differentiable`21_in z0 : ( ( ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like )
Element of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ) holds
SVF1 (1 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) ,
(pdiff1 (f : ( ( V21() ) ( V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( (
V21()
quasi_total ) ( non
empty V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
total quasi_total complex-valued ext-real-valued real-valued )
Function of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ,
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) ,
z0 : ( ( ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like )
Element of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ) ) : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ,
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
is_continuous_in (proj (1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ,2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( (
V21()
quasi_total ) ( non
empty V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
total quasi_total complex-valued ext-real-valued real-valued )
Element of
K6(
K7(
(REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ,
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
. z0 : ( ( ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like )
Element of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) ;
theorem
for
f being ( (
V21() ) (
V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
for
z0 being ( ( ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like )
Element of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ) st
f : ( (
V21() ) (
V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
is_hpartial_differentiable`22_in z0 : ( ( ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like )
Element of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ) holds
SVF1 (2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) ,
(pdiff1 (f : ( ( V21() ) ( V16() V19( REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) M11( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) )) ) V20( REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) V21() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( (
V21()
quasi_total ) ( non
empty V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
total quasi_total complex-valued ext-real-valued real-valued )
Function of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ,
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) ,
z0 : ( ( ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like )
Element of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ) ) : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ,
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
is_continuous_in (proj (2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ,2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( (
V21()
quasi_total ) ( non
empty V16()
V19(
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
total quasi_total complex-valued ext-real-valued real-valued )
Element of
K6(
K7(
(REAL 2 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V136() V159() V160() V161() V162() V163() V164() V165() ) Element of NAT : ( ( ) ( V160() V161() V162() V163() V164() V165() V166() ) Element of K6(REAL : ( ( ) ( non empty V54() V160() V161() V162() V166() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ,
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
. z0 : ( ( ) (
V16()
V19(
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )
V21()
complex-valued ext-real-valued real-valued V61(2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like )
Element of
REAL 2 : ( ( ) ( non
empty ordinal natural V11()
real ext-real positive non
negative V136()
V159()
V160()
V161()
V162()
V163()
V164()
V165() )
Element of
NAT : ( ( ) (
V160()
V161()
V162()
V163()
V164()
V165()
V166() )
Element of
K6(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
empty FinSequence-membered )
M11(
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) )) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V54()
V160()
V161()
V162()
V166() )
set ) ) ;