begin
theorem
for
f being ( (
Function-like ) (
Relation-like COMPLEX : ( ( ) ( non
zero V33()
V166()
V172() )
set )
-defined COMPLEX : ( ( ) ( non
zero V33()
V166()
V172() )
set )
-valued Function-like complex-yielding )
PartFunc of ,)
for
u,
v being ( (
Function-like ) (
Relation-like REAL 2 : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive non
negative V126()
V127()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
zero FinSequence-membered )
M14(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ))
-defined REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set )
-valued Function-like complex-yielding V115()
V116() )
PartFunc of ,)
for
z0 being ( (
V11() ) (
V11() )
Complex)
for
x0,
y0 being ( ( ) (
V11()
real ext-real )
Real)
for
xy0 being ( ( ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set )
-valued Function-like V40(2 : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive non
negative V126()
V127()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like complex-yielding V115()
V116() )
Element of
REAL 2 : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive non
negative V126()
V127()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
zero FinSequence-membered )
M14(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) )) ) st ( for
x,
y being ( ( ) (
V11()
real ext-real )
Real) st
x : ( ( ) (
V11()
real ext-real )
Real)
+ (y : ( ( ) ( V11() real ext-real ) Real) * <i> : ( ( ) ( V11() ) Element of COMPLEX : ( ( ) ( non zero V33() V166() V172() ) set ) ) ) : ( ( ) (
V11() )
Element of
COMPLEX : ( ( ) ( non
zero V33()
V166()
V172() )
set ) ) : ( ( ) (
V11() )
Element of
COMPLEX : ( ( ) ( non
zero V33()
V166()
V172() )
set ) )
in dom f : ( (
Function-like ) (
Relation-like COMPLEX : ( ( ) ( non
zero V33()
V166()
V172() )
set )
-defined COMPLEX : ( ( ) ( non
zero V33()
V166()
V172() )
set )
-valued Function-like complex-yielding )
PartFunc of ,) : ( ( ) (
V166() )
Element of
K6(
COMPLEX : ( ( ) ( non
zero V33()
V166()
V172() )
set ) ) : ( ( ) ( )
set ) ) holds
(
<*x : ( ( ) ( V11() real ext-real ) Real) ,y : ( ( ) ( V11() real ext-real ) Real) *> : ( ( ) ( )
set )
in dom u : ( (
Function-like ) (
Relation-like REAL 2 : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive non
negative V126()
V127()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
zero FinSequence-membered )
M14(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ))
-defined REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set )
-valued Function-like complex-yielding V115()
V116() )
PartFunc of ,) : ( ( ) ( )
Element of
K6(
(REAL 2 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V126() V127() V166() V167() V168() V169() V170() V171() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V166() V167() V168() V169() V170() V171() V172() ) Element of K6(REAL : ( ( ) ( non zero V33() V166() V167() V168() V172() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non
zero FinSequence-membered )
M14(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) )) ) : ( ( ) ( )
set ) ) &
u : ( (
Function-like ) (
Relation-like REAL 2 : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive non
negative V126()
V127()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
zero FinSequence-membered )
M14(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ))
-defined REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set )
-valued Function-like complex-yielding V115()
V116() )
PartFunc of ,)
. <*x : ( ( ) ( V11() real ext-real ) Real) ,y : ( ( ) ( V11() real ext-real ) Real) *> : ( ( ) ( )
set ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) )
= (Re f : ( ( Function-like ) ( Relation-like COMPLEX : ( ( ) ( non zero V33() V166() V172() ) set ) -defined COMPLEX : ( ( ) ( non zero V33() V166() V172() ) set ) -valued Function-like complex-yielding ) PartFunc of ,) ) : ( (
Function-like ) (
Relation-like COMPLEX : ( ( ) ( non
zero V33()
V166()
V172() )
set )
-defined REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set )
-valued Function-like complex-yielding V115()
V116() )
Element of
K6(
K7(
COMPLEX : ( ( ) ( non
zero V33()
V166()
V172() )
set ) ,
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) (
complex-yielding V115()
V116() )
set ) ) : ( ( ) ( )
set ) )
. (x : ( ( ) ( V11() real ext-real ) Real) + (y : ( ( ) ( V11() real ext-real ) Real) * <i> : ( ( ) ( V11() ) Element of COMPLEX : ( ( ) ( non zero V33() V166() V172() ) set ) ) ) : ( ( ) ( V11() ) Element of COMPLEX : ( ( ) ( non zero V33() V166() V172() ) set ) ) ) : ( ( ) (
V11() )
Element of
COMPLEX : ( ( ) ( non
zero V33()
V166()
V172() )
set ) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) ) ) & ( for
x,
y being ( ( ) (
V11()
real ext-real )
Real) st
x : ( ( ) (
V11()
real ext-real )
Real)
+ (y : ( ( ) ( V11() real ext-real ) Real) * <i> : ( ( ) ( V11() ) Element of COMPLEX : ( ( ) ( non zero V33() V166() V172() ) set ) ) ) : ( ( ) (
V11() )
Element of
COMPLEX : ( ( ) ( non
zero V33()
V166()
V172() )
set ) ) : ( ( ) (
V11() )
Element of
COMPLEX : ( ( ) ( non
zero V33()
V166()
V172() )
set ) )
in dom f : ( (
Function-like ) (
Relation-like COMPLEX : ( ( ) ( non
zero V33()
V166()
V172() )
set )
-defined COMPLEX : ( ( ) ( non
zero V33()
V166()
V172() )
set )
-valued Function-like complex-yielding )
PartFunc of ,) : ( ( ) (
V166() )
Element of
K6(
COMPLEX : ( ( ) ( non
zero V33()
V166()
V172() )
set ) ) : ( ( ) ( )
set ) ) holds
(
<*x : ( ( ) ( V11() real ext-real ) Real) ,y : ( ( ) ( V11() real ext-real ) Real) *> : ( ( ) ( )
set )
in dom v : ( (
Function-like ) (
Relation-like REAL 2 : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive non
negative V126()
V127()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
zero FinSequence-membered )
M14(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ))
-defined REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set )
-valued Function-like complex-yielding V115()
V116() )
PartFunc of ,) : ( ( ) ( )
Element of
K6(
(REAL 2 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V126() V127() V166() V167() V168() V169() V170() V171() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V166() V167() V168() V169() V170() V171() V172() ) Element of K6(REAL : ( ( ) ( non zero V33() V166() V167() V168() V172() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non
zero FinSequence-membered )
M14(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) )) ) : ( ( ) ( )
set ) ) &
v : ( (
Function-like ) (
Relation-like REAL 2 : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive non
negative V126()
V127()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
zero FinSequence-membered )
M14(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ))
-defined REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set )
-valued Function-like complex-yielding V115()
V116() )
PartFunc of ,)
. <*x : ( ( ) ( V11() real ext-real ) Real) ,y : ( ( ) ( V11() real ext-real ) Real) *> : ( ( ) ( )
set ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) )
= (Im f : ( ( Function-like ) ( Relation-like COMPLEX : ( ( ) ( non zero V33() V166() V172() ) set ) -defined COMPLEX : ( ( ) ( non zero V33() V166() V172() ) set ) -valued Function-like complex-yielding ) PartFunc of ,) ) : ( (
Function-like ) (
Relation-like COMPLEX : ( ( ) ( non
zero V33()
V166()
V172() )
set )
-defined REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set )
-valued Function-like complex-yielding V115()
V116() )
Element of
K6(
K7(
COMPLEX : ( ( ) ( non
zero V33()
V166()
V172() )
set ) ,
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) (
complex-yielding V115()
V116() )
set ) ) : ( ( ) ( )
set ) )
. (x : ( ( ) ( V11() real ext-real ) Real) + (y : ( ( ) ( V11() real ext-real ) Real) * <i> : ( ( ) ( V11() ) Element of COMPLEX : ( ( ) ( non zero V33() V166() V172() ) set ) ) ) : ( ( ) ( V11() ) Element of COMPLEX : ( ( ) ( non zero V33() V166() V172() ) set ) ) ) : ( ( ) (
V11() )
Element of
COMPLEX : ( ( ) ( non
zero V33()
V166()
V172() )
set ) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) ) ) &
z0 : ( (
V11() ) (
V11() )
Complex)
= x0 : ( ( ) (
V11()
real ext-real )
Real)
+ (y0 : ( ( ) ( V11() real ext-real ) Real) * <i> : ( ( ) ( V11() ) Element of COMPLEX : ( ( ) ( non zero V33() V166() V172() ) set ) ) ) : ( ( ) (
V11() )
Element of
COMPLEX : ( ( ) ( non
zero V33()
V166()
V172() )
set ) ) : ( ( ) (
V11() )
Element of
COMPLEX : ( ( ) ( non
zero V33()
V166()
V172() )
set ) ) &
xy0 : ( ( ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set )
-valued Function-like V40(2 : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive non
negative V126()
V127()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like complex-yielding V115()
V116() )
Element of
REAL 2 : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive non
negative V126()
V127()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
zero FinSequence-membered )
M14(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) )) )
= <*x0 : ( ( ) ( V11() real ext-real ) Real) ,y0 : ( ( ) ( V11() real ext-real ) Real) *> : ( ( ) ( )
set ) &
f : ( (
Function-like ) (
Relation-like COMPLEX : ( ( ) ( non
zero V33()
V166()
V172() )
set )
-defined COMPLEX : ( ( ) ( non
zero V33()
V166()
V172() )
set )
-valued Function-like complex-yielding )
PartFunc of ,)
is_differentiable_in z0 : ( (
V11() ) (
V11() )
Complex) holds
(
u : ( (
Function-like ) (
Relation-like REAL 2 : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive non
negative V126()
V127()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
zero FinSequence-membered )
M14(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ))
-defined REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set )
-valued Function-like complex-yielding V115()
V116() )
PartFunc of ,)
is_partial_differentiable_in xy0 : ( ( ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set )
-valued Function-like V40(2 : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive non
negative V126()
V127()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like complex-yielding V115()
V116() )
Element of
REAL 2 : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive non
negative V126()
V127()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
zero FinSequence-membered )
M14(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) )) ) ,1 : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive non
negative V126()
V127()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) &
u : ( (
Function-like ) (
Relation-like REAL 2 : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive non
negative V126()
V127()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
zero FinSequence-membered )
M14(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ))
-defined REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set )
-valued Function-like complex-yielding V115()
V116() )
PartFunc of ,)
is_partial_differentiable_in xy0 : ( ( ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set )
-valued Function-like V40(2 : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive non
negative V126()
V127()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like complex-yielding V115()
V116() )
Element of
REAL 2 : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive non
negative V126()
V127()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
zero FinSequence-membered )
M14(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) )) ) ,2 : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive non
negative V126()
V127()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) &
v : ( (
Function-like ) (
Relation-like REAL 2 : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive non
negative V126()
V127()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
zero FinSequence-membered )
M14(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ))
-defined REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set )
-valued Function-like complex-yielding V115()
V116() )
PartFunc of ,)
is_partial_differentiable_in xy0 : ( ( ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set )
-valued Function-like V40(2 : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive non
negative V126()
V127()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like complex-yielding V115()
V116() )
Element of
REAL 2 : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive non
negative V126()
V127()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
zero FinSequence-membered )
M14(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) )) ) ,1 : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive non
negative V126()
V127()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) &
v : ( (
Function-like ) (
Relation-like REAL 2 : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive non
negative V126()
V127()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
zero FinSequence-membered )
M14(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ))
-defined REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set )
-valued Function-like complex-yielding V115()
V116() )
PartFunc of ,)
is_partial_differentiable_in xy0 : ( ( ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set )
-valued Function-like V40(2 : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive non
negative V126()
V127()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like complex-yielding V115()
V116() )
Element of
REAL 2 : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive non
negative V126()
V127()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
zero FinSequence-membered )
M14(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) )) ) ,2 : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive non
negative V126()
V127()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) &
Re (diff (f : ( ( Function-like ) ( Relation-like COMPLEX : ( ( ) ( non zero V33() V166() V172() ) set ) -defined COMPLEX : ( ( ) ( non zero V33() V166() V172() ) set ) -valued Function-like complex-yielding ) PartFunc of ,) ,z0 : ( ( V11() ) ( V11() ) Complex) )) : ( ( ) (
V11() )
Element of
COMPLEX : ( ( ) ( non
zero V33()
V166()
V172() )
set ) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) )
= partdiff (
u : ( (
Function-like ) (
Relation-like REAL 2 : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive non
negative V126()
V127()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
zero FinSequence-membered )
M14(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ))
-defined REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set )
-valued Function-like complex-yielding V115()
V116() )
PartFunc of ,) ,
xy0 : ( ( ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set )
-valued Function-like V40(2 : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive non
negative V126()
V127()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like complex-yielding V115()
V116() )
Element of
REAL 2 : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive non
negative V126()
V127()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
zero FinSequence-membered )
M14(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) )) ) ,1 : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive non
negative V126()
V127()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) &
Re (diff (f : ( ( Function-like ) ( Relation-like COMPLEX : ( ( ) ( non zero V33() V166() V172() ) set ) -defined COMPLEX : ( ( ) ( non zero V33() V166() V172() ) set ) -valued Function-like complex-yielding ) PartFunc of ,) ,z0 : ( ( V11() ) ( V11() ) Complex) )) : ( ( ) (
V11() )
Element of
COMPLEX : ( ( ) ( non
zero V33()
V166()
V172() )
set ) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) )
= partdiff (
v : ( (
Function-like ) (
Relation-like REAL 2 : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive non
negative V126()
V127()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
zero FinSequence-membered )
M14(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ))
-defined REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set )
-valued Function-like complex-yielding V115()
V116() )
PartFunc of ,) ,
xy0 : ( ( ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set )
-valued Function-like V40(2 : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive non
negative V126()
V127()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like complex-yielding V115()
V116() )
Element of
REAL 2 : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive non
negative V126()
V127()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
zero FinSequence-membered )
M14(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) )) ) ,2 : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive non
negative V126()
V127()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) &
Im (diff (f : ( ( Function-like ) ( Relation-like COMPLEX : ( ( ) ( non zero V33() V166() V172() ) set ) -defined COMPLEX : ( ( ) ( non zero V33() V166() V172() ) set ) -valued Function-like complex-yielding ) PartFunc of ,) ,z0 : ( ( V11() ) ( V11() ) Complex) )) : ( ( ) (
V11() )
Element of
COMPLEX : ( ( ) ( non
zero V33()
V166()
V172() )
set ) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) )
= - (partdiff (u : ( ( Function-like ) ( Relation-like REAL 2 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V126() V127() V166() V167() V168() V169() V170() V171() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V166() V167() V168() V169() V170() V171() V172() ) Element of K6(REAL : ( ( ) ( non zero V33() V166() V167() V168() V172() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non zero FinSequence-membered ) M14( REAL : ( ( ) ( non zero V33() V166() V167() V168() V172() ) set ) )) -defined REAL : ( ( ) ( non zero V33() V166() V167() V168() V172() ) set ) -valued Function-like complex-yielding V115() V116() ) PartFunc of ,) ,xy0 : ( ( ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V166() V167() V168() V169() V170() V171() V172() ) Element of K6(REAL : ( ( ) ( non zero V33() V166() V167() V168() V172() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V33() V166() V167() V168() V172() ) set ) -valued Function-like V40(2 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V126() V127() V166() V167() V168() V169() V170() V171() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V166() V167() V168() V169() V170() V171() V172() ) Element of K6(REAL : ( ( ) ( non zero V33() V166() V167() V168() V172() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V115() V116() ) Element of REAL 2 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V126() V127() V166() V167() V168() V169() V170() V171() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V166() V167() V168() V169() V170() V171() V172() ) Element of K6(REAL : ( ( ) ( non zero V33() V166() V167() V168() V172() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non zero FinSequence-membered ) M14( REAL : ( ( ) ( non zero V33() V166() V167() V168() V172() ) set ) )) ) ,2 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V126() V127() V166() V167() V168() V169() V170() V171() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V166() V167() V168() V169() V170() V171() V172() ) Element of K6(REAL : ( ( ) ( non zero V33() V166() V167() V168() V172() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) &
Im (diff (f : ( ( Function-like ) ( Relation-like COMPLEX : ( ( ) ( non zero V33() V166() V172() ) set ) -defined COMPLEX : ( ( ) ( non zero V33() V166() V172() ) set ) -valued Function-like complex-yielding ) PartFunc of ,) ,z0 : ( ( V11() ) ( V11() ) Complex) )) : ( ( ) (
V11() )
Element of
COMPLEX : ( ( ) ( non
zero V33()
V166()
V172() )
set ) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) )
= partdiff (
v : ( (
Function-like ) (
Relation-like REAL 2 : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive non
negative V126()
V127()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
zero FinSequence-membered )
M14(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ))
-defined REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set )
-valued Function-like complex-yielding V115()
V116() )
PartFunc of ,) ,
xy0 : ( ( ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set )
-valued Function-like V40(2 : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive non
negative V126()
V127()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like complex-yielding V115()
V116() )
Element of
REAL 2 : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive non
negative V126()
V127()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
zero FinSequence-membered )
M14(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) )) ) ,1 : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive non
negative V126()
V127()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) ) ;
theorem
for
u being ( (
Function-like ) (
Relation-like REAL 2 : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive non
negative V126()
V127()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
zero FinSequence-membered )
M14(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ))
-defined REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set )
-valued Function-like complex-yielding V115()
V116() )
PartFunc of ,)
for
x0,
y0 being ( ( ) (
V11()
real ext-real )
Real)
for
xy0 being ( ( ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set )
-valued Function-like V40(2 : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive non
negative V126()
V127()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like complex-yielding V115()
V116() )
Element of
REAL 2 : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive non
negative V126()
V127()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
zero FinSequence-membered )
M14(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) )) ) st
xy0 : ( ( ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set )
-valued Function-like V40(2 : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive non
negative V126()
V127()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like complex-yielding V115()
V116() )
Element of
REAL 2 : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive non
negative V126()
V127()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
zero FinSequence-membered )
M14(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) )) )
= <*x0 : ( ( ) ( V11() real ext-real ) Real) ,y0 : ( ( ) ( V11() real ext-real ) Real) *> : ( ( ) ( )
set ) &
<>* u : ( (
Function-like ) (
Relation-like REAL 2 : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive non
negative V126()
V127()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
zero FinSequence-membered )
M14(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ))
-defined REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set )
-valued Function-like complex-yielding V115()
V116() )
PartFunc of ,) : ( (
Function-like ) (
Relation-like REAL 2 : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive non
negative V126()
V127()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
zero FinSequence-membered )
M14(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ))
-defined REAL 1 : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive non
negative V126()
V127()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
zero FinSequence-membered )
M14(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ))
-valued Function-like )
Element of
K6(
K7(
(REAL 2 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V126() V127() V166() V167() V168() V169() V170() V171() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V166() V167() V168() V169() V170() V171() V172() ) Element of K6(REAL : ( ( ) ( non zero V33() V166() V167() V168() V172() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non
zero FinSequence-membered )
M14(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) )) ,
(REAL 1 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V126() V127() V166() V167() V168() V169() V170() V171() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V166() V167() V168() V169() V170() V171() V172() ) Element of K6(REAL : ( ( ) ( non zero V33() V166() V167() V168() V172() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non
zero FinSequence-membered )
M14(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) )) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_in xy0 : ( ( ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set )
-valued Function-like V40(2 : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive non
negative V126()
V127()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like complex-yielding V115()
V116() )
Element of
REAL 2 : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive non
negative V126()
V127()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
zero FinSequence-membered )
M14(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) )) ) holds
(
u : ( (
Function-like ) (
Relation-like REAL 2 : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive non
negative V126()
V127()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
zero FinSequence-membered )
M14(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ))
-defined REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set )
-valued Function-like complex-yielding V115()
V116() )
PartFunc of ,)
is_partial_differentiable_in xy0 : ( ( ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set )
-valued Function-like V40(2 : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive non
negative V126()
V127()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like complex-yielding V115()
V116() )
Element of
REAL 2 : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive non
negative V126()
V127()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
zero FinSequence-membered )
M14(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) )) ) ,1 : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive non
negative V126()
V127()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) &
u : ( (
Function-like ) (
Relation-like REAL 2 : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive non
negative V126()
V127()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
zero FinSequence-membered )
M14(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ))
-defined REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set )
-valued Function-like complex-yielding V115()
V116() )
PartFunc of ,)
is_partial_differentiable_in xy0 : ( ( ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set )
-valued Function-like V40(2 : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive non
negative V126()
V127()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like complex-yielding V115()
V116() )
Element of
REAL 2 : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive non
negative V126()
V127()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
zero FinSequence-membered )
M14(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) )) ) ,2 : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive non
negative V126()
V127()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) &
<*(partdiff (u : ( ( Function-like ) ( Relation-like REAL 2 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V126() V127() V166() V167() V168() V169() V170() V171() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V166() V167() V168() V169() V170() V171() V172() ) Element of K6(REAL : ( ( ) ( non zero V33() V166() V167() V168() V172() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non zero FinSequence-membered ) M14( REAL : ( ( ) ( non zero V33() V166() V167() V168() V172() ) set ) )) -defined REAL : ( ( ) ( non zero V33() V166() V167() V168() V172() ) set ) -valued Function-like complex-yielding V115() V116() ) PartFunc of ,) ,xy0 : ( ( ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V166() V167() V168() V169() V170() V171() V172() ) Element of K6(REAL : ( ( ) ( non zero V33() V166() V167() V168() V172() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V33() V166() V167() V168() V172() ) set ) -valued Function-like V40(2 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V126() V127() V166() V167() V168() V169() V170() V171() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V166() V167() V168() V169() V170() V171() V172() ) Element of K6(REAL : ( ( ) ( non zero V33() V166() V167() V168() V172() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V115() V116() ) Element of REAL 2 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V126() V127() V166() V167() V168() V169() V170() V171() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V166() V167() V168() V169() V170() V171() V172() ) Element of K6(REAL : ( ( ) ( non zero V33() V166() V167() V168() V172() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non zero FinSequence-membered ) M14( REAL : ( ( ) ( non zero V33() V166() V167() V168() V172() ) set ) )) ) ,1 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V126() V127() V166() V167() V168() V169() V170() V171() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V166() V167() V168() V169() V170() V171() V172() ) Element of K6(REAL : ( ( ) ( non zero V33() V166() V167() V168() V172() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non zero V33() V166() V167() V168() V172() ) set ) ) *> : ( ( ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set )
-valued Function-like FinSequence-like complex-yielding V115()
V116() )
FinSequence of
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) )
= (diff ((<>* u : ( ( Function-like ) ( Relation-like REAL 2 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V126() V127() V166() V167() V168() V169() V170() V171() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V166() V167() V168() V169() V170() V171() V172() ) Element of K6(REAL : ( ( ) ( non zero V33() V166() V167() V168() V172() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non zero FinSequence-membered ) M14( REAL : ( ( ) ( non zero V33() V166() V167() V168() V172() ) set ) )) -defined REAL : ( ( ) ( non zero V33() V166() V167() V168() V172() ) set ) -valued Function-like complex-yielding V115() V116() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like REAL 2 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V126() V127() V166() V167() V168() V169() V170() V171() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V166() V167() V168() V169() V170() V171() V172() ) Element of K6(REAL : ( ( ) ( non zero V33() V166() V167() V168() V172() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non zero FinSequence-membered ) M14( REAL : ( ( ) ( non zero V33() V166() V167() V168() V172() ) set ) )) -defined REAL 1 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V126() V127() V166() V167() V168() V169() V170() V171() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V166() V167() V168() V169() V170() V171() V172() ) Element of K6(REAL : ( ( ) ( non zero V33() V166() V167() V168() V172() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non zero FinSequence-membered ) M14( REAL : ( ( ) ( non zero V33() V166() V167() V168() V172() ) set ) )) -valued Function-like ) Element of K6(K7((REAL 2 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V126() V127() V166() V167() V168() V169() V170() V171() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V166() V167() V168() V169() V170() V171() V172() ) Element of K6(REAL : ( ( ) ( non zero V33() V166() V167() V168() V172() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non zero FinSequence-membered ) M14( REAL : ( ( ) ( non zero V33() V166() V167() V168() V172() ) set ) )) ,(REAL 1 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V126() V127() V166() V167() V168() V169() V170() V171() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V166() V167() V168() V169() V170() V171() V172() ) Element of K6(REAL : ( ( ) ( non zero V33() V166() V167() V168() V172() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non zero FinSequence-membered ) M14( REAL : ( ( ) ( non zero V33() V166() V167() V168() V172() ) set ) )) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,xy0 : ( ( ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V166() V167() V168() V169() V170() V171() V172() ) Element of K6(REAL : ( ( ) ( non zero V33() V166() V167() V168() V172() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V33() V166() V167() V168() V172() ) set ) -valued Function-like V40(2 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V126() V127() V166() V167() V168() V169() V170() V171() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V166() V167() V168() V169() V170() V171() V172() ) Element of K6(REAL : ( ( ) ( non zero V33() V166() V167() V168() V172() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V115() V116() ) Element of REAL 2 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V126() V127() V166() V167() V168() V169() V170() V171() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V166() V167() V168() V169() V170() V171() V172() ) Element of K6(REAL : ( ( ) ( non zero V33() V166() V167() V168() V172() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non zero FinSequence-membered ) M14( REAL : ( ( ) ( non zero V33() V166() V167() V168() V172() ) set ) )) ) )) : ( (
Function-like quasi_total ) ( non
zero Relation-like REAL 2 : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive non
negative V126()
V127()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
zero FinSequence-membered )
M14(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ))
-defined REAL 1 : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive non
negative V126()
V127()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
zero FinSequence-membered )
M14(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ))
-valued Function-like total quasi_total )
Element of
K6(
K7(
(REAL 2 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V126() V127() V166() V167() V168() V169() V170() V171() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V166() V167() V168() V169() V170() V171() V172() ) Element of K6(REAL : ( ( ) ( non zero V33() V166() V167() V168() V172() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non
zero FinSequence-membered )
M14(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) )) ,
(REAL 1 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V126() V127() V166() V167() V168() V169() V170() V171() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V166() V167() V168() V169() V170() V171() V172() ) Element of K6(REAL : ( ( ) ( non zero V33() V166() V167() V168() V172() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non
zero FinSequence-membered )
M14(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) )) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
. <*1 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V126() V127() V166() V167() V168() V169() V170() V171() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V166() V167() V168() V169() V170() V171() V172() ) Element of K6(REAL : ( ( ) ( non zero V33() V166() V167() V168() V172() ) set ) ) : ( ( ) ( ) set ) ) ) ,0 : ( ( ) ( zero epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative V126() V127() V166() V167() V168() V169() V170() V171() V172() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V166() V167() V168() V169() V170() V171() V172() ) Element of K6(REAL : ( ( ) ( non zero V33() V166() V167() V168() V172() ) set ) ) : ( ( ) ( ) set ) ) ) *> : ( ( ) ( )
set ) : ( ( ) ( )
set ) &
<*(partdiff (u : ( ( Function-like ) ( Relation-like REAL 2 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V126() V127() V166() V167() V168() V169() V170() V171() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V166() V167() V168() V169() V170() V171() V172() ) Element of K6(REAL : ( ( ) ( non zero V33() V166() V167() V168() V172() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non zero FinSequence-membered ) M14( REAL : ( ( ) ( non zero V33() V166() V167() V168() V172() ) set ) )) -defined REAL : ( ( ) ( non zero V33() V166() V167() V168() V172() ) set ) -valued Function-like complex-yielding V115() V116() ) PartFunc of ,) ,xy0 : ( ( ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V166() V167() V168() V169() V170() V171() V172() ) Element of K6(REAL : ( ( ) ( non zero V33() V166() V167() V168() V172() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V33() V166() V167() V168() V172() ) set ) -valued Function-like V40(2 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V126() V127() V166() V167() V168() V169() V170() V171() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V166() V167() V168() V169() V170() V171() V172() ) Element of K6(REAL : ( ( ) ( non zero V33() V166() V167() V168() V172() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V115() V116() ) Element of REAL 2 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V126() V127() V166() V167() V168() V169() V170() V171() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V166() V167() V168() V169() V170() V171() V172() ) Element of K6(REAL : ( ( ) ( non zero V33() V166() V167() V168() V172() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non zero FinSequence-membered ) M14( REAL : ( ( ) ( non zero V33() V166() V167() V168() V172() ) set ) )) ) ,2 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V126() V127() V166() V167() V168() V169() V170() V171() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V166() V167() V168() V169() V170() V171() V172() ) Element of K6(REAL : ( ( ) ( non zero V33() V166() V167() V168() V172() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non zero V33() V166() V167() V168() V172() ) set ) ) *> : ( ( ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set )
-valued Function-like FinSequence-like complex-yielding V115()
V116() )
FinSequence of
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) )
= (diff ((<>* u : ( ( Function-like ) ( Relation-like REAL 2 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V126() V127() V166() V167() V168() V169() V170() V171() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V166() V167() V168() V169() V170() V171() V172() ) Element of K6(REAL : ( ( ) ( non zero V33() V166() V167() V168() V172() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non zero FinSequence-membered ) M14( REAL : ( ( ) ( non zero V33() V166() V167() V168() V172() ) set ) )) -defined REAL : ( ( ) ( non zero V33() V166() V167() V168() V172() ) set ) -valued Function-like complex-yielding V115() V116() ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like REAL 2 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V126() V127() V166() V167() V168() V169() V170() V171() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V166() V167() V168() V169() V170() V171() V172() ) Element of K6(REAL : ( ( ) ( non zero V33() V166() V167() V168() V172() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non zero FinSequence-membered ) M14( REAL : ( ( ) ( non zero V33() V166() V167() V168() V172() ) set ) )) -defined REAL 1 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V126() V127() V166() V167() V168() V169() V170() V171() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V166() V167() V168() V169() V170() V171() V172() ) Element of K6(REAL : ( ( ) ( non zero V33() V166() V167() V168() V172() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non zero FinSequence-membered ) M14( REAL : ( ( ) ( non zero V33() V166() V167() V168() V172() ) set ) )) -valued Function-like ) Element of K6(K7((REAL 2 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V126() V127() V166() V167() V168() V169() V170() V171() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V166() V167() V168() V169() V170() V171() V172() ) Element of K6(REAL : ( ( ) ( non zero V33() V166() V167() V168() V172() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non zero FinSequence-membered ) M14( REAL : ( ( ) ( non zero V33() V166() V167() V168() V172() ) set ) )) ,(REAL 1 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V126() V127() V166() V167() V168() V169() V170() V171() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V166() V167() V168() V169() V170() V171() V172() ) Element of K6(REAL : ( ( ) ( non zero V33() V166() V167() V168() V172() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non zero FinSequence-membered ) M14( REAL : ( ( ) ( non zero V33() V166() V167() V168() V172() ) set ) )) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,xy0 : ( ( ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V166() V167() V168() V169() V170() V171() V172() ) Element of K6(REAL : ( ( ) ( non zero V33() V166() V167() V168() V172() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V33() V166() V167() V168() V172() ) set ) -valued Function-like V40(2 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V126() V127() V166() V167() V168() V169() V170() V171() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V166() V167() V168() V169() V170() V171() V172() ) Element of K6(REAL : ( ( ) ( non zero V33() V166() V167() V168() V172() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V115() V116() ) Element of REAL 2 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V126() V127() V166() V167() V168() V169() V170() V171() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V166() V167() V168() V169() V170() V171() V172() ) Element of K6(REAL : ( ( ) ( non zero V33() V166() V167() V168() V172() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non zero FinSequence-membered ) M14( REAL : ( ( ) ( non zero V33() V166() V167() V168() V172() ) set ) )) ) )) : ( (
Function-like quasi_total ) ( non
zero Relation-like REAL 2 : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive non
negative V126()
V127()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
zero FinSequence-membered )
M14(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ))
-defined REAL 1 : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive non
negative V126()
V127()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
zero FinSequence-membered )
M14(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ))
-valued Function-like total quasi_total )
Element of
K6(
K7(
(REAL 2 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V126() V127() V166() V167() V168() V169() V170() V171() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V166() V167() V168() V169() V170() V171() V172() ) Element of K6(REAL : ( ( ) ( non zero V33() V166() V167() V168() V172() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non
zero FinSequence-membered )
M14(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) )) ,
(REAL 1 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V126() V127() V166() V167() V168() V169() V170() V171() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V166() V167() V168() V169() V170() V171() V172() ) Element of K6(REAL : ( ( ) ( non zero V33() V166() V167() V168() V172() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non
zero FinSequence-membered )
M14(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) )) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
. <*0 : ( ( ) ( zero epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative V126() V127() V166() V167() V168() V169() V170() V171() V172() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V166() V167() V168() V169() V170() V171() V172() ) Element of K6(REAL : ( ( ) ( non zero V33() V166() V167() V168() V172() ) set ) ) : ( ( ) ( ) set ) ) ) ,1 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V126() V127() V166() V167() V168() V169() V170() V171() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V166() V167() V168() V169() V170() V171() V172() ) Element of K6(REAL : ( ( ) ( non zero V33() V166() V167() V168() V172() ) set ) ) : ( ( ) ( ) set ) ) ) *> : ( ( ) ( )
set ) : ( ( ) ( )
set ) ) ;
theorem
for
f being ( (
Function-like ) (
Relation-like COMPLEX : ( ( ) ( non
zero V33()
V166()
V172() )
set )
-defined COMPLEX : ( ( ) ( non
zero V33()
V166()
V172() )
set )
-valued Function-like complex-yielding )
PartFunc of ,)
for
u,
v being ( (
Function-like ) (
Relation-like REAL 2 : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive non
negative V126()
V127()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
zero FinSequence-membered )
M14(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ))
-defined REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set )
-valued Function-like complex-yielding V115()
V116() )
PartFunc of ,)
for
z0 being ( (
V11() ) (
V11() )
Complex)
for
x0,
y0 being ( ( ) (
V11()
real ext-real )
Real)
for
xy0 being ( ( ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set )
-valued Function-like V40(2 : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive non
negative V126()
V127()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like complex-yielding V115()
V116() )
Element of
REAL 2 : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive non
negative V126()
V127()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
zero FinSequence-membered )
M14(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) )) ) st ( for
x,
y being ( ( ) (
V11()
real ext-real )
Real) st
<*x : ( ( ) ( V11() real ext-real ) Real) ,y : ( ( ) ( V11() real ext-real ) Real) *> : ( ( ) ( )
set )
in dom v : ( (
Function-like ) (
Relation-like REAL 2 : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive non
negative V126()
V127()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
zero FinSequence-membered )
M14(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ))
-defined REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set )
-valued Function-like complex-yielding V115()
V116() )
PartFunc of ,) : ( ( ) ( )
Element of
K6(
(REAL 2 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V126() V127() V166() V167() V168() V169() V170() V171() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V166() V167() V168() V169() V170() V171() V172() ) Element of K6(REAL : ( ( ) ( non zero V33() V166() V167() V168() V172() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non
zero FinSequence-membered )
M14(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) )) ) : ( ( ) ( )
set ) ) holds
x : ( ( ) (
V11()
real ext-real )
Real)
+ (y : ( ( ) ( V11() real ext-real ) Real) * <i> : ( ( ) ( V11() ) Element of COMPLEX : ( ( ) ( non zero V33() V166() V172() ) set ) ) ) : ( ( ) (
V11() )
Element of
COMPLEX : ( ( ) ( non
zero V33()
V166()
V172() )
set ) ) : ( ( ) (
V11() )
Element of
COMPLEX : ( ( ) ( non
zero V33()
V166()
V172() )
set ) )
in dom f : ( (
Function-like ) (
Relation-like COMPLEX : ( ( ) ( non
zero V33()
V166()
V172() )
set )
-defined COMPLEX : ( ( ) ( non
zero V33()
V166()
V172() )
set )
-valued Function-like complex-yielding )
PartFunc of ,) : ( ( ) (
V166() )
Element of
K6(
COMPLEX : ( ( ) ( non
zero V33()
V166()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) & ( for
x,
y being ( ( ) (
V11()
real ext-real )
Real) st
x : ( ( ) (
V11()
real ext-real )
Real)
+ (y : ( ( ) ( V11() real ext-real ) Real) * <i> : ( ( ) ( V11() ) Element of COMPLEX : ( ( ) ( non zero V33() V166() V172() ) set ) ) ) : ( ( ) (
V11() )
Element of
COMPLEX : ( ( ) ( non
zero V33()
V166()
V172() )
set ) ) : ( ( ) (
V11() )
Element of
COMPLEX : ( ( ) ( non
zero V33()
V166()
V172() )
set ) )
in dom f : ( (
Function-like ) (
Relation-like COMPLEX : ( ( ) ( non
zero V33()
V166()
V172() )
set )
-defined COMPLEX : ( ( ) ( non
zero V33()
V166()
V172() )
set )
-valued Function-like complex-yielding )
PartFunc of ,) : ( ( ) (
V166() )
Element of
K6(
COMPLEX : ( ( ) ( non
zero V33()
V166()
V172() )
set ) ) : ( ( ) ( )
set ) ) holds
(
<*x : ( ( ) ( V11() real ext-real ) Real) ,y : ( ( ) ( V11() real ext-real ) Real) *> : ( ( ) ( )
set )
in dom u : ( (
Function-like ) (
Relation-like REAL 2 : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive non
negative V126()
V127()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
zero FinSequence-membered )
M14(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ))
-defined REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set )
-valued Function-like complex-yielding V115()
V116() )
PartFunc of ,) : ( ( ) ( )
Element of
K6(
(REAL 2 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V126() V127() V166() V167() V168() V169() V170() V171() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V166() V167() V168() V169() V170() V171() V172() ) Element of K6(REAL : ( ( ) ( non zero V33() V166() V167() V168() V172() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non
zero FinSequence-membered )
M14(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) )) ) : ( ( ) ( )
set ) ) &
u : ( (
Function-like ) (
Relation-like REAL 2 : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive non
negative V126()
V127()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
zero FinSequence-membered )
M14(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ))
-defined REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set )
-valued Function-like complex-yielding V115()
V116() )
PartFunc of ,)
. <*x : ( ( ) ( V11() real ext-real ) Real) ,y : ( ( ) ( V11() real ext-real ) Real) *> : ( ( ) ( )
set ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) )
= (Re f : ( ( Function-like ) ( Relation-like COMPLEX : ( ( ) ( non zero V33() V166() V172() ) set ) -defined COMPLEX : ( ( ) ( non zero V33() V166() V172() ) set ) -valued Function-like complex-yielding ) PartFunc of ,) ) : ( (
Function-like ) (
Relation-like COMPLEX : ( ( ) ( non
zero V33()
V166()
V172() )
set )
-defined REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set )
-valued Function-like complex-yielding V115()
V116() )
Element of
K6(
K7(
COMPLEX : ( ( ) ( non
zero V33()
V166()
V172() )
set ) ,
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) (
complex-yielding V115()
V116() )
set ) ) : ( ( ) ( )
set ) )
. (x : ( ( ) ( V11() real ext-real ) Real) + (y : ( ( ) ( V11() real ext-real ) Real) * <i> : ( ( ) ( V11() ) Element of COMPLEX : ( ( ) ( non zero V33() V166() V172() ) set ) ) ) : ( ( ) ( V11() ) Element of COMPLEX : ( ( ) ( non zero V33() V166() V172() ) set ) ) ) : ( ( ) (
V11() )
Element of
COMPLEX : ( ( ) ( non
zero V33()
V166()
V172() )
set ) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) ) ) & ( for
x,
y being ( ( ) (
V11()
real ext-real )
Real) st
x : ( ( ) (
V11()
real ext-real )
Real)
+ (y : ( ( ) ( V11() real ext-real ) Real) * <i> : ( ( ) ( V11() ) Element of COMPLEX : ( ( ) ( non zero V33() V166() V172() ) set ) ) ) : ( ( ) (
V11() )
Element of
COMPLEX : ( ( ) ( non
zero V33()
V166()
V172() )
set ) ) : ( ( ) (
V11() )
Element of
COMPLEX : ( ( ) ( non
zero V33()
V166()
V172() )
set ) )
in dom f : ( (
Function-like ) (
Relation-like COMPLEX : ( ( ) ( non
zero V33()
V166()
V172() )
set )
-defined COMPLEX : ( ( ) ( non
zero V33()
V166()
V172() )
set )
-valued Function-like complex-yielding )
PartFunc of ,) : ( ( ) (
V166() )
Element of
K6(
COMPLEX : ( ( ) ( non
zero V33()
V166()
V172() )
set ) ) : ( ( ) ( )
set ) ) holds
(
<*x : ( ( ) ( V11() real ext-real ) Real) ,y : ( ( ) ( V11() real ext-real ) Real) *> : ( ( ) ( )
set )
in dom v : ( (
Function-like ) (
Relation-like REAL 2 : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive non
negative V126()
V127()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
zero FinSequence-membered )
M14(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ))
-defined REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set )
-valued Function-like complex-yielding V115()
V116() )
PartFunc of ,) : ( ( ) ( )
Element of
K6(
(REAL 2 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V126() V127() V166() V167() V168() V169() V170() V171() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V166() V167() V168() V169() V170() V171() V172() ) Element of K6(REAL : ( ( ) ( non zero V33() V166() V167() V168() V172() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non
zero FinSequence-membered )
M14(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) )) ) : ( ( ) ( )
set ) ) &
v : ( (
Function-like ) (
Relation-like REAL 2 : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive non
negative V126()
V127()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
zero FinSequence-membered )
M14(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ))
-defined REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set )
-valued Function-like complex-yielding V115()
V116() )
PartFunc of ,)
. <*x : ( ( ) ( V11() real ext-real ) Real) ,y : ( ( ) ( V11() real ext-real ) Real) *> : ( ( ) ( )
set ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) )
= (Im f : ( ( Function-like ) ( Relation-like COMPLEX : ( ( ) ( non zero V33() V166() V172() ) set ) -defined COMPLEX : ( ( ) ( non zero V33() V166() V172() ) set ) -valued Function-like complex-yielding ) PartFunc of ,) ) : ( (
Function-like ) (
Relation-like COMPLEX : ( ( ) ( non
zero V33()
V166()
V172() )
set )
-defined REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set )
-valued Function-like complex-yielding V115()
V116() )
Element of
K6(
K7(
COMPLEX : ( ( ) ( non
zero V33()
V166()
V172() )
set ) ,
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) (
complex-yielding V115()
V116() )
set ) ) : ( ( ) ( )
set ) )
. (x : ( ( ) ( V11() real ext-real ) Real) + (y : ( ( ) ( V11() real ext-real ) Real) * <i> : ( ( ) ( V11() ) Element of COMPLEX : ( ( ) ( non zero V33() V166() V172() ) set ) ) ) : ( ( ) ( V11() ) Element of COMPLEX : ( ( ) ( non zero V33() V166() V172() ) set ) ) ) : ( ( ) (
V11() )
Element of
COMPLEX : ( ( ) ( non
zero V33()
V166()
V172() )
set ) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) ) ) &
z0 : ( (
V11() ) (
V11() )
Complex)
= x0 : ( ( ) (
V11()
real ext-real )
Real)
+ (y0 : ( ( ) ( V11() real ext-real ) Real) * <i> : ( ( ) ( V11() ) Element of COMPLEX : ( ( ) ( non zero V33() V166() V172() ) set ) ) ) : ( ( ) (
V11() )
Element of
COMPLEX : ( ( ) ( non
zero V33()
V166()
V172() )
set ) ) : ( ( ) (
V11() )
Element of
COMPLEX : ( ( ) ( non
zero V33()
V166()
V172() )
set ) ) &
xy0 : ( ( ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set )
-valued Function-like V40(2 : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive non
negative V126()
V127()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like complex-yielding V115()
V116() )
Element of
REAL 2 : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive non
negative V126()
V127()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
zero FinSequence-membered )
M14(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) )) )
= <*x0 : ( ( ) ( V11() real ext-real ) Real) ,y0 : ( ( ) ( V11() real ext-real ) Real) *> : ( ( ) ( )
set ) &
<>* u : ( (
Function-like ) (
Relation-like REAL 2 : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive non
negative V126()
V127()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
zero FinSequence-membered )
M14(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ))
-defined REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set )
-valued Function-like complex-yielding V115()
V116() )
PartFunc of ,) : ( (
Function-like ) (
Relation-like REAL 2 : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive non
negative V126()
V127()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
zero FinSequence-membered )
M14(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ))
-defined REAL 1 : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive non
negative V126()
V127()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
zero FinSequence-membered )
M14(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ))
-valued Function-like )
Element of
K6(
K7(
(REAL 2 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V126() V127() V166() V167() V168() V169() V170() V171() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V166() V167() V168() V169() V170() V171() V172() ) Element of K6(REAL : ( ( ) ( non zero V33() V166() V167() V168() V172() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non
zero FinSequence-membered )
M14(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) )) ,
(REAL 1 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V126() V127() V166() V167() V168() V169() V170() V171() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V166() V167() V168() V169() V170() V171() V172() ) Element of K6(REAL : ( ( ) ( non zero V33() V166() V167() V168() V172() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non
zero FinSequence-membered )
M14(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) )) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_in xy0 : ( ( ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set )
-valued Function-like V40(2 : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive non
negative V126()
V127()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like complex-yielding V115()
V116() )
Element of
REAL 2 : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive non
negative V126()
V127()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
zero FinSequence-membered )
M14(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) )) ) &
<>* v : ( (
Function-like ) (
Relation-like REAL 2 : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive non
negative V126()
V127()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
zero FinSequence-membered )
M14(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ))
-defined REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set )
-valued Function-like complex-yielding V115()
V116() )
PartFunc of ,) : ( (
Function-like ) (
Relation-like REAL 2 : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive non
negative V126()
V127()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
zero FinSequence-membered )
M14(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ))
-defined REAL 1 : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive non
negative V126()
V127()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
zero FinSequence-membered )
M14(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ))
-valued Function-like )
Element of
K6(
K7(
(REAL 2 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V126() V127() V166() V167() V168() V169() V170() V171() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V166() V167() V168() V169() V170() V171() V172() ) Element of K6(REAL : ( ( ) ( non zero V33() V166() V167() V168() V172() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non
zero FinSequence-membered )
M14(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) )) ,
(REAL 1 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V126() V127() V166() V167() V168() V169() V170() V171() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V166() V167() V168() V169() V170() V171() V172() ) Element of K6(REAL : ( ( ) ( non zero V33() V166() V167() V168() V172() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non
zero FinSequence-membered )
M14(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) )) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_in xy0 : ( ( ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set )
-valued Function-like V40(2 : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive non
negative V126()
V127()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like complex-yielding V115()
V116() )
Element of
REAL 2 : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive non
negative V126()
V127()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
zero FinSequence-membered )
M14(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) )) ) &
partdiff (
u : ( (
Function-like ) (
Relation-like REAL 2 : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive non
negative V126()
V127()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
zero FinSequence-membered )
M14(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ))
-defined REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set )
-valued Function-like complex-yielding V115()
V116() )
PartFunc of ,) ,
xy0 : ( ( ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set )
-valued Function-like V40(2 : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive non
negative V126()
V127()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like complex-yielding V115()
V116() )
Element of
REAL 2 : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive non
negative V126()
V127()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
zero FinSequence-membered )
M14(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) )) ) ,1 : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive non
negative V126()
V127()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) )
= partdiff (
v : ( (
Function-like ) (
Relation-like REAL 2 : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive non
negative V126()
V127()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
zero FinSequence-membered )
M14(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ))
-defined REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set )
-valued Function-like complex-yielding V115()
V116() )
PartFunc of ,) ,
xy0 : ( ( ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set )
-valued Function-like V40(2 : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive non
negative V126()
V127()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like complex-yielding V115()
V116() )
Element of
REAL 2 : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive non
negative V126()
V127()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
zero FinSequence-membered )
M14(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) )) ) ,2 : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive non
negative V126()
V127()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) &
partdiff (
u : ( (
Function-like ) (
Relation-like REAL 2 : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive non
negative V126()
V127()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
zero FinSequence-membered )
M14(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ))
-defined REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set )
-valued Function-like complex-yielding V115()
V116() )
PartFunc of ,) ,
xy0 : ( ( ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set )
-valued Function-like V40(2 : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive non
negative V126()
V127()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like complex-yielding V115()
V116() )
Element of
REAL 2 : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive non
negative V126()
V127()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
zero FinSequence-membered )
M14(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) )) ) ,2 : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive non
negative V126()
V127()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) )
= - (partdiff (v : ( ( Function-like ) ( Relation-like REAL 2 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V126() V127() V166() V167() V168() V169() V170() V171() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V166() V167() V168() V169() V170() V171() V172() ) Element of K6(REAL : ( ( ) ( non zero V33() V166() V167() V168() V172() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non zero FinSequence-membered ) M14( REAL : ( ( ) ( non zero V33() V166() V167() V168() V172() ) set ) )) -defined REAL : ( ( ) ( non zero V33() V166() V167() V168() V172() ) set ) -valued Function-like complex-yielding V115() V116() ) PartFunc of ,) ,xy0 : ( ( ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V166() V167() V168() V169() V170() V171() V172() ) Element of K6(REAL : ( ( ) ( non zero V33() V166() V167() V168() V172() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V33() V166() V167() V168() V172() ) set ) -valued Function-like V40(2 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V126() V127() V166() V167() V168() V169() V170() V171() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V166() V167() V168() V169() V170() V171() V172() ) Element of K6(REAL : ( ( ) ( non zero V33() V166() V167() V168() V172() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V115() V116() ) Element of REAL 2 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V126() V127() V166() V167() V168() V169() V170() V171() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V166() V167() V168() V169() V170() V171() V172() ) Element of K6(REAL : ( ( ) ( non zero V33() V166() V167() V168() V172() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non zero FinSequence-membered ) M14( REAL : ( ( ) ( non zero V33() V166() V167() V168() V172() ) set ) )) ) ,1 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V126() V127() V166() V167() V168() V169() V170() V171() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V166() V167() V168() V169() V170() V171() V172() ) Element of K6(REAL : ( ( ) ( non zero V33() V166() V167() V168() V172() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) holds
(
f : ( (
Function-like ) (
Relation-like COMPLEX : ( ( ) ( non
zero V33()
V166()
V172() )
set )
-defined COMPLEX : ( ( ) ( non
zero V33()
V166()
V172() )
set )
-valued Function-like complex-yielding )
PartFunc of ,)
is_differentiable_in z0 : ( (
V11() ) (
V11() )
Complex) &
u : ( (
Function-like ) (
Relation-like REAL 2 : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive non
negative V126()
V127()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
zero FinSequence-membered )
M14(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ))
-defined REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set )
-valued Function-like complex-yielding V115()
V116() )
PartFunc of ,)
is_partial_differentiable_in xy0 : ( ( ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set )
-valued Function-like V40(2 : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive non
negative V126()
V127()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like complex-yielding V115()
V116() )
Element of
REAL 2 : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive non
negative V126()
V127()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
zero FinSequence-membered )
M14(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) )) ) ,1 : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive non
negative V126()
V127()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) &
u : ( (
Function-like ) (
Relation-like REAL 2 : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive non
negative V126()
V127()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
zero FinSequence-membered )
M14(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ))
-defined REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set )
-valued Function-like complex-yielding V115()
V116() )
PartFunc of ,)
is_partial_differentiable_in xy0 : ( ( ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set )
-valued Function-like V40(2 : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive non
negative V126()
V127()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like complex-yielding V115()
V116() )
Element of
REAL 2 : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive non
negative V126()
V127()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
zero FinSequence-membered )
M14(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) )) ) ,2 : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive non
negative V126()
V127()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) &
v : ( (
Function-like ) (
Relation-like REAL 2 : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive non
negative V126()
V127()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
zero FinSequence-membered )
M14(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ))
-defined REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set )
-valued Function-like complex-yielding V115()
V116() )
PartFunc of ,)
is_partial_differentiable_in xy0 : ( ( ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set )
-valued Function-like V40(2 : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive non
negative V126()
V127()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like complex-yielding V115()
V116() )
Element of
REAL 2 : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive non
negative V126()
V127()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
zero FinSequence-membered )
M14(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) )) ) ,1 : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive non
negative V126()
V127()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) &
v : ( (
Function-like ) (
Relation-like REAL 2 : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive non
negative V126()
V127()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
zero FinSequence-membered )
M14(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ))
-defined REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set )
-valued Function-like complex-yielding V115()
V116() )
PartFunc of ,)
is_partial_differentiable_in xy0 : ( ( ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set )
-valued Function-like V40(2 : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive non
negative V126()
V127()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like complex-yielding V115()
V116() )
Element of
REAL 2 : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive non
negative V126()
V127()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
zero FinSequence-membered )
M14(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) )) ) ,2 : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive non
negative V126()
V127()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) &
Re (diff (f : ( ( Function-like ) ( Relation-like COMPLEX : ( ( ) ( non zero V33() V166() V172() ) set ) -defined COMPLEX : ( ( ) ( non zero V33() V166() V172() ) set ) -valued Function-like complex-yielding ) PartFunc of ,) ,z0 : ( ( V11() ) ( V11() ) Complex) )) : ( ( ) (
V11() )
Element of
COMPLEX : ( ( ) ( non
zero V33()
V166()
V172() )
set ) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) )
= partdiff (
u : ( (
Function-like ) (
Relation-like REAL 2 : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive non
negative V126()
V127()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
zero FinSequence-membered )
M14(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ))
-defined REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set )
-valued Function-like complex-yielding V115()
V116() )
PartFunc of ,) ,
xy0 : ( ( ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set )
-valued Function-like V40(2 : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive non
negative V126()
V127()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like complex-yielding V115()
V116() )
Element of
REAL 2 : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive non
negative V126()
V127()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
zero FinSequence-membered )
M14(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) )) ) ,1 : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive non
negative V126()
V127()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) &
Re (diff (f : ( ( Function-like ) ( Relation-like COMPLEX : ( ( ) ( non zero V33() V166() V172() ) set ) -defined COMPLEX : ( ( ) ( non zero V33() V166() V172() ) set ) -valued Function-like complex-yielding ) PartFunc of ,) ,z0 : ( ( V11() ) ( V11() ) Complex) )) : ( ( ) (
V11() )
Element of
COMPLEX : ( ( ) ( non
zero V33()
V166()
V172() )
set ) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) )
= partdiff (
v : ( (
Function-like ) (
Relation-like REAL 2 : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive non
negative V126()
V127()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
zero FinSequence-membered )
M14(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ))
-defined REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set )
-valued Function-like complex-yielding V115()
V116() )
PartFunc of ,) ,
xy0 : ( ( ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set )
-valued Function-like V40(2 : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive non
negative V126()
V127()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like complex-yielding V115()
V116() )
Element of
REAL 2 : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive non
negative V126()
V127()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
zero FinSequence-membered )
M14(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) )) ) ,2 : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive non
negative V126()
V127()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) &
Im (diff (f : ( ( Function-like ) ( Relation-like COMPLEX : ( ( ) ( non zero V33() V166() V172() ) set ) -defined COMPLEX : ( ( ) ( non zero V33() V166() V172() ) set ) -valued Function-like complex-yielding ) PartFunc of ,) ,z0 : ( ( V11() ) ( V11() ) Complex) )) : ( ( ) (
V11() )
Element of
COMPLEX : ( ( ) ( non
zero V33()
V166()
V172() )
set ) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) )
= - (partdiff (u : ( ( Function-like ) ( Relation-like REAL 2 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V126() V127() V166() V167() V168() V169() V170() V171() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V166() V167() V168() V169() V170() V171() V172() ) Element of K6(REAL : ( ( ) ( non zero V33() V166() V167() V168() V172() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non zero FinSequence-membered ) M14( REAL : ( ( ) ( non zero V33() V166() V167() V168() V172() ) set ) )) -defined REAL : ( ( ) ( non zero V33() V166() V167() V168() V172() ) set ) -valued Function-like complex-yielding V115() V116() ) PartFunc of ,) ,xy0 : ( ( ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V166() V167() V168() V169() V170() V171() V172() ) Element of K6(REAL : ( ( ) ( non zero V33() V166() V167() V168() V172() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V33() V166() V167() V168() V172() ) set ) -valued Function-like V40(2 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V126() V127() V166() V167() V168() V169() V170() V171() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V166() V167() V168() V169() V170() V171() V172() ) Element of K6(REAL : ( ( ) ( non zero V33() V166() V167() V168() V172() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-yielding V115() V116() ) Element of REAL 2 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V126() V127() V166() V167() V168() V169() V170() V171() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V166() V167() V168() V169() V170() V171() V172() ) Element of K6(REAL : ( ( ) ( non zero V33() V166() V167() V168() V172() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non zero FinSequence-membered ) M14( REAL : ( ( ) ( non zero V33() V166() V167() V168() V172() ) set ) )) ) ,2 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V126() V127() V166() V167() V168() V169() V170() V171() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V166() V167() V168() V169() V170() V171() V172() ) Element of K6(REAL : ( ( ) ( non zero V33() V166() V167() V168() V172() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) &
Im (diff (f : ( ( Function-like ) ( Relation-like COMPLEX : ( ( ) ( non zero V33() V166() V172() ) set ) -defined COMPLEX : ( ( ) ( non zero V33() V166() V172() ) set ) -valued Function-like complex-yielding ) PartFunc of ,) ,z0 : ( ( V11() ) ( V11() ) Complex) )) : ( ( ) (
V11() )
Element of
COMPLEX : ( ( ) ( non
zero V33()
V166()
V172() )
set ) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) )
= partdiff (
v : ( (
Function-like ) (
Relation-like REAL 2 : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive non
negative V126()
V127()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
zero FinSequence-membered )
M14(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ))
-defined REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set )
-valued Function-like complex-yielding V115()
V116() )
PartFunc of ,) ,
xy0 : ( ( ) (
Relation-like NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) )
-defined REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set )
-valued Function-like V40(2 : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive non
negative V126()
V127()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) )
FinSequence-like complex-yielding V115()
V116() )
Element of
REAL 2 : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive non
negative V126()
V127()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( non
zero FinSequence-membered )
M14(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) )) ) ,1 : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal natural V11()
real ext-real positive non
negative V126()
V127()
V166()
V167()
V168()
V169()
V170()
V171() )
Element of
NAT : ( ( ) ( non
zero epsilon-transitive epsilon-connected ordinal V166()
V167()
V168()
V169()
V170()
V171()
V172() )
Element of
K6(
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) : ( ( ) ( )
set ) ) ) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) ( non
zero V33()
V166()
V167()
V168()
V172() )
set ) ) ) ;