begin
theorem
for
x0 being ( ( ) (
V30()
real ext-real )
Real)
for
f1,
f2 being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,) st
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
is_left_divergent_to+infty_in x0 : ( ( ) (
V30()
real ext-real )
Real) &
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
is_left_divergent_to+infty_in x0 : ( ( ) (
V30()
real ext-real )
Real) & ( for
r being ( ( ) (
V30()
real ext-real )
Real) st
r : ( ( ) (
V30()
real ext-real )
Real)
< x0 : ( ( ) (
V30()
real ext-real )
Real) holds
ex
g being ( ( ) (
V30()
real ext-real )
Real) st
(
r : ( ( ) (
V30()
real ext-real )
Real)
< g : ( ( ) (
V30()
real ext-real )
Real) &
g : ( ( ) (
V30()
real ext-real )
Real)
< x0 : ( ( ) (
V30()
real ext-real )
Real) &
g : ( ( ) (
V30()
real ext-real )
Real)
in (dom f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) )
/\ (dom f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) ) ) holds
(
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
+ f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
is_left_divergent_to+infty_in x0 : ( ( ) (
V30()
real ext-real )
Real) &
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
(#) f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
is_left_divergent_to+infty_in x0 : ( ( ) (
V30()
real ext-real )
Real) ) ;
theorem
for
x0 being ( ( ) (
V30()
real ext-real )
Real)
for
f1,
f2 being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,) st
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
is_left_divergent_to-infty_in x0 : ( ( ) (
V30()
real ext-real )
Real) &
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
is_left_divergent_to-infty_in x0 : ( ( ) (
V30()
real ext-real )
Real) & ( for
r being ( ( ) (
V30()
real ext-real )
Real) st
r : ( ( ) (
V30()
real ext-real )
Real)
< x0 : ( ( ) (
V30()
real ext-real )
Real) holds
ex
g being ( ( ) (
V30()
real ext-real )
Real) st
(
r : ( ( ) (
V30()
real ext-real )
Real)
< g : ( ( ) (
V30()
real ext-real )
Real) &
g : ( ( ) (
V30()
real ext-real )
Real)
< x0 : ( ( ) (
V30()
real ext-real )
Real) &
g : ( ( ) (
V30()
real ext-real )
Real)
in (dom f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) )
/\ (dom f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) ) ) holds
(
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
+ f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
is_left_divergent_to-infty_in x0 : ( ( ) (
V30()
real ext-real )
Real) &
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
(#) f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
is_left_divergent_to+infty_in x0 : ( ( ) (
V30()
real ext-real )
Real) ) ;
theorem
for
x0 being ( ( ) (
V30()
real ext-real )
Real)
for
f1,
f2 being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,) st
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
is_right_divergent_to+infty_in x0 : ( ( ) (
V30()
real ext-real )
Real) &
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
is_right_divergent_to+infty_in x0 : ( ( ) (
V30()
real ext-real )
Real) & ( for
r being ( ( ) (
V30()
real ext-real )
Real) st
x0 : ( ( ) (
V30()
real ext-real )
Real)
< r : ( ( ) (
V30()
real ext-real )
Real) holds
ex
g being ( ( ) (
V30()
real ext-real )
Real) st
(
g : ( ( ) (
V30()
real ext-real )
Real)
< r : ( ( ) (
V30()
real ext-real )
Real) &
x0 : ( ( ) (
V30()
real ext-real )
Real)
< g : ( ( ) (
V30()
real ext-real )
Real) &
g : ( ( ) (
V30()
real ext-real )
Real)
in (dom f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) )
/\ (dom f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) ) ) holds
(
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
+ f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
is_right_divergent_to+infty_in x0 : ( ( ) (
V30()
real ext-real )
Real) &
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
(#) f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
is_right_divergent_to+infty_in x0 : ( ( ) (
V30()
real ext-real )
Real) ) ;
theorem
for
x0 being ( ( ) (
V30()
real ext-real )
Real)
for
f1,
f2 being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,) st
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
is_right_divergent_to-infty_in x0 : ( ( ) (
V30()
real ext-real )
Real) &
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
is_right_divergent_to-infty_in x0 : ( ( ) (
V30()
real ext-real )
Real) & ( for
r being ( ( ) (
V30()
real ext-real )
Real) st
x0 : ( ( ) (
V30()
real ext-real )
Real)
< r : ( ( ) (
V30()
real ext-real )
Real) holds
ex
g being ( ( ) (
V30()
real ext-real )
Real) st
(
g : ( ( ) (
V30()
real ext-real )
Real)
< r : ( ( ) (
V30()
real ext-real )
Real) &
x0 : ( ( ) (
V30()
real ext-real )
Real)
< g : ( ( ) (
V30()
real ext-real )
Real) &
g : ( ( ) (
V30()
real ext-real )
Real)
in (dom f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) )
/\ (dom f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) ) ) holds
(
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
+ f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
is_right_divergent_to-infty_in x0 : ( ( ) (
V30()
real ext-real )
Real) &
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
(#) f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
is_right_divergent_to+infty_in x0 : ( ( ) (
V30()
real ext-real )
Real) ) ;
theorem
for
x0 being ( ( ) (
V30()
real ext-real )
Real)
for
f1,
f2 being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,) st
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
is_left_divergent_to+infty_in x0 : ( ( ) (
V30()
real ext-real )
Real) & ( for
r being ( ( ) (
V30()
real ext-real )
Real) st
r : ( ( ) (
V30()
real ext-real )
Real)
< x0 : ( ( ) (
V30()
real ext-real )
Real) holds
ex
g being ( ( ) (
V30()
real ext-real )
Real) st
(
r : ( ( ) (
V30()
real ext-real )
Real)
< g : ( ( ) (
V30()
real ext-real )
Real) &
g : ( ( ) (
V30()
real ext-real )
Real)
< x0 : ( ( ) (
V30()
real ext-real )
Real) &
g : ( ( ) (
V30()
real ext-real )
Real)
in dom (f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) + f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) ) ) & ex
r being ( ( ) (
V30()
real ext-real )
Real) st
(
0 : ( ( ) (
empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V30()
real ext-real non
positive non
negative V71()
V72()
V73()
V74()
V75()
V76()
V77()
V87()
V88() )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) )
< r : ( ( ) (
V30()
real ext-real )
Real) &
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
| ].(x0 : ( ( ) ( V30() real ext-real ) Real) - r : ( ( ) ( V30() real ext-real ) Real) ) : ( ( ) ( V30() real ext-real ) Element of REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) ,x0 : ( ( ) ( V30() real ext-real ) Real) .[ : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) is
bounded_below ) holds
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
+ f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
is_left_divergent_to+infty_in x0 : ( ( ) (
V30()
real ext-real )
Real) ;
theorem
for
x0 being ( ( ) (
V30()
real ext-real )
Real)
for
f1,
f2 being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,) st
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
is_left_divergent_to+infty_in x0 : ( ( ) (
V30()
real ext-real )
Real) & ( for
r being ( ( ) (
V30()
real ext-real )
Real) st
r : ( ( ) (
V30()
real ext-real )
Real)
< x0 : ( ( ) (
V30()
real ext-real )
Real) holds
ex
g being ( ( ) (
V30()
real ext-real )
Real) st
(
r : ( ( ) (
V30()
real ext-real )
Real)
< g : ( ( ) (
V30()
real ext-real )
Real) &
g : ( ( ) (
V30()
real ext-real )
Real)
< x0 : ( ( ) (
V30()
real ext-real )
Real) &
g : ( ( ) (
V30()
real ext-real )
Real)
in dom (f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) (#) f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) ) ) & ex
r,
r1 being ( ( ) (
V30()
real ext-real )
Real) st
(
0 : ( ( ) (
empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V30()
real ext-real non
positive non
negative V71()
V72()
V73()
V74()
V75()
V76()
V77()
V87()
V88() )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) )
< r : ( ( ) (
V30()
real ext-real )
Real) &
0 : ( ( ) (
empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V30()
real ext-real non
positive non
negative V71()
V72()
V73()
V74()
V75()
V76()
V77()
V87()
V88() )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) )
< r1 : ( ( ) (
V30()
real ext-real )
Real) & ( for
g being ( ( ) (
V30()
real ext-real )
Real) st
g : ( ( ) (
V30()
real ext-real )
Real)
in (dom f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) )
/\ ].(x0 : ( ( ) ( V30() real ext-real ) Real) - r : ( ( ) ( V30() real ext-real ) Real) ) : ( ( ) ( V30() real ext-real ) Element of REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) ,x0 : ( ( ) ( V30() real ext-real ) Real) .[ : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) holds
r1 : ( ( ) (
V30()
real ext-real )
Real)
<= f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
. g : ( ( ) (
V30()
real ext-real )
Real) : ( ( ) (
V30()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) ) ) holds
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
(#) f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
is_left_divergent_to+infty_in x0 : ( ( ) (
V30()
real ext-real )
Real) ;
theorem
for
x0 being ( ( ) (
V30()
real ext-real )
Real)
for
f1,
f2 being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,) st
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
is_right_divergent_to+infty_in x0 : ( ( ) (
V30()
real ext-real )
Real) & ( for
r being ( ( ) (
V30()
real ext-real )
Real) st
x0 : ( ( ) (
V30()
real ext-real )
Real)
< r : ( ( ) (
V30()
real ext-real )
Real) holds
ex
g being ( ( ) (
V30()
real ext-real )
Real) st
(
g : ( ( ) (
V30()
real ext-real )
Real)
< r : ( ( ) (
V30()
real ext-real )
Real) &
x0 : ( ( ) (
V30()
real ext-real )
Real)
< g : ( ( ) (
V30()
real ext-real )
Real) &
g : ( ( ) (
V30()
real ext-real )
Real)
in dom (f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) + f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) ) ) & ex
r being ( ( ) (
V30()
real ext-real )
Real) st
(
0 : ( ( ) (
empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V30()
real ext-real non
positive non
negative V71()
V72()
V73()
V74()
V75()
V76()
V77()
V87()
V88() )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) )
< r : ( ( ) (
V30()
real ext-real )
Real) &
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
| ].x0 : ( ( ) ( V30() real ext-real ) Real) ,(x0 : ( ( ) ( V30() real ext-real ) Real) + r : ( ( ) ( V30() real ext-real ) Real) ) : ( ( ) ( V30() real ext-real ) Element of REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) .[ : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) is
bounded_below ) holds
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
+ f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
is_right_divergent_to+infty_in x0 : ( ( ) (
V30()
real ext-real )
Real) ;
theorem
for
x0 being ( ( ) (
V30()
real ext-real )
Real)
for
f1,
f2 being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,) st
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
is_right_divergent_to+infty_in x0 : ( ( ) (
V30()
real ext-real )
Real) & ( for
r being ( ( ) (
V30()
real ext-real )
Real) st
x0 : ( ( ) (
V30()
real ext-real )
Real)
< r : ( ( ) (
V30()
real ext-real )
Real) holds
ex
g being ( ( ) (
V30()
real ext-real )
Real) st
(
g : ( ( ) (
V30()
real ext-real )
Real)
< r : ( ( ) (
V30()
real ext-real )
Real) &
x0 : ( ( ) (
V30()
real ext-real )
Real)
< g : ( ( ) (
V30()
real ext-real )
Real) &
g : ( ( ) (
V30()
real ext-real )
Real)
in dom (f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) (#) f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) ) ) & ex
r,
r1 being ( ( ) (
V30()
real ext-real )
Real) st
(
0 : ( ( ) (
empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V30()
real ext-real non
positive non
negative V71()
V72()
V73()
V74()
V75()
V76()
V77()
V87()
V88() )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) )
< r : ( ( ) (
V30()
real ext-real )
Real) &
0 : ( ( ) (
empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V30()
real ext-real non
positive non
negative V71()
V72()
V73()
V74()
V75()
V76()
V77()
V87()
V88() )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) )
< r1 : ( ( ) (
V30()
real ext-real )
Real) & ( for
g being ( ( ) (
V30()
real ext-real )
Real) st
g : ( ( ) (
V30()
real ext-real )
Real)
in (dom f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) )
/\ ].x0 : ( ( ) ( V30() real ext-real ) Real) ,(x0 : ( ( ) ( V30() real ext-real ) Real) + r : ( ( ) ( V30() real ext-real ) Real) ) : ( ( ) ( V30() real ext-real ) Element of REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) .[ : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) holds
r1 : ( ( ) (
V30()
real ext-real )
Real)
<= f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
. g : ( ( ) (
V30()
real ext-real )
Real) : ( ( ) (
V30()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) ) ) holds
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
(#) f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
is_right_divergent_to+infty_in x0 : ( ( ) (
V30()
real ext-real )
Real) ;
theorem
for
x0,
r being ( ( ) (
V30()
real ext-real )
Real)
for
f being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,) holds
( (
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
is_left_divergent_to+infty_in x0 : ( ( ) (
V30()
real ext-real )
Real) &
r : ( ( ) (
V30()
real ext-real )
Real)
> 0 : ( ( ) (
empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V30()
real ext-real non
positive non
negative V71()
V72()
V73()
V74()
V75()
V76()
V77()
V87()
V88() )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) ) implies
r : ( ( ) (
V30()
real ext-real )
Real)
(#) f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
is_left_divergent_to+infty_in x0 : ( ( ) (
V30()
real ext-real )
Real) ) & (
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
is_left_divergent_to+infty_in x0 : ( ( ) (
V30()
real ext-real )
Real) &
r : ( ( ) (
V30()
real ext-real )
Real)
< 0 : ( ( ) (
empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V30()
real ext-real non
positive non
negative V71()
V72()
V73()
V74()
V75()
V76()
V77()
V87()
V88() )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) ) implies
r : ( ( ) (
V30()
real ext-real )
Real)
(#) f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
is_left_divergent_to-infty_in x0 : ( ( ) (
V30()
real ext-real )
Real) ) & (
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
is_left_divergent_to-infty_in x0 : ( ( ) (
V30()
real ext-real )
Real) &
r : ( ( ) (
V30()
real ext-real )
Real)
> 0 : ( ( ) (
empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V30()
real ext-real non
positive non
negative V71()
V72()
V73()
V74()
V75()
V76()
V77()
V87()
V88() )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) ) implies
r : ( ( ) (
V30()
real ext-real )
Real)
(#) f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
is_left_divergent_to-infty_in x0 : ( ( ) (
V30()
real ext-real )
Real) ) & (
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
is_left_divergent_to-infty_in x0 : ( ( ) (
V30()
real ext-real )
Real) &
r : ( ( ) (
V30()
real ext-real )
Real)
< 0 : ( ( ) (
empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V30()
real ext-real non
positive non
negative V71()
V72()
V73()
V74()
V75()
V76()
V77()
V87()
V88() )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) ) implies
r : ( ( ) (
V30()
real ext-real )
Real)
(#) f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
is_left_divergent_to+infty_in x0 : ( ( ) (
V30()
real ext-real )
Real) ) ) ;
theorem
for
x0,
r being ( ( ) (
V30()
real ext-real )
Real)
for
f being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,) holds
( (
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
is_right_divergent_to+infty_in x0 : ( ( ) (
V30()
real ext-real )
Real) &
r : ( ( ) (
V30()
real ext-real )
Real)
> 0 : ( ( ) (
empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V30()
real ext-real non
positive non
negative V71()
V72()
V73()
V74()
V75()
V76()
V77()
V87()
V88() )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) ) implies
r : ( ( ) (
V30()
real ext-real )
Real)
(#) f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
is_right_divergent_to+infty_in x0 : ( ( ) (
V30()
real ext-real )
Real) ) & (
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
is_right_divergent_to+infty_in x0 : ( ( ) (
V30()
real ext-real )
Real) &
r : ( ( ) (
V30()
real ext-real )
Real)
< 0 : ( ( ) (
empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V30()
real ext-real non
positive non
negative V71()
V72()
V73()
V74()
V75()
V76()
V77()
V87()
V88() )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) ) implies
r : ( ( ) (
V30()
real ext-real )
Real)
(#) f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
is_right_divergent_to-infty_in x0 : ( ( ) (
V30()
real ext-real )
Real) ) & (
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
is_right_divergent_to-infty_in x0 : ( ( ) (
V30()
real ext-real )
Real) &
r : ( ( ) (
V30()
real ext-real )
Real)
> 0 : ( ( ) (
empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V30()
real ext-real non
positive non
negative V71()
V72()
V73()
V74()
V75()
V76()
V77()
V87()
V88() )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) ) implies
r : ( ( ) (
V30()
real ext-real )
Real)
(#) f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
is_right_divergent_to-infty_in x0 : ( ( ) (
V30()
real ext-real )
Real) ) & (
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
is_right_divergent_to-infty_in x0 : ( ( ) (
V30()
real ext-real )
Real) &
r : ( ( ) (
V30()
real ext-real )
Real)
< 0 : ( ( ) (
empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V30()
real ext-real non
positive non
negative V71()
V72()
V73()
V74()
V75()
V76()
V77()
V87()
V88() )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) ) implies
r : ( ( ) (
V30()
real ext-real )
Real)
(#) f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
is_right_divergent_to+infty_in x0 : ( ( ) (
V30()
real ext-real )
Real) ) ) ;
theorem
for
x0 being ( ( ) (
V30()
real ext-real )
Real)
for
f1,
f being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,) st
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
is_left_divergent_to+infty_in x0 : ( ( ) (
V30()
real ext-real )
Real) & ( for
r being ( ( ) (
V30()
real ext-real )
Real) st
r : ( ( ) (
V30()
real ext-real )
Real)
< x0 : ( ( ) (
V30()
real ext-real )
Real) holds
ex
g being ( ( ) (
V30()
real ext-real )
Real) st
(
r : ( ( ) (
V30()
real ext-real )
Real)
< g : ( ( ) (
V30()
real ext-real )
Real) &
g : ( ( ) (
V30()
real ext-real )
Real)
< x0 : ( ( ) (
V30()
real ext-real )
Real) &
g : ( ( ) (
V30()
real ext-real )
Real)
in dom f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) ) ) & ex
r being ( ( ) (
V30()
real ext-real )
Real) st
(
0 : ( ( ) (
empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V30()
real ext-real non
positive non
negative V71()
V72()
V73()
V74()
V75()
V76()
V77()
V87()
V88() )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) )
< r : ( ( ) (
V30()
real ext-real )
Real) &
(dom f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) )
/\ ].(x0 : ( ( ) ( V30() real ext-real ) Real) - r : ( ( ) ( V30() real ext-real ) Real) ) : ( ( ) ( V30() real ext-real ) Element of REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) ,x0 : ( ( ) ( V30() real ext-real ) Real) .[ : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) )
c= (dom f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) )
/\ ].(x0 : ( ( ) ( V30() real ext-real ) Real) - r : ( ( ) ( V30() real ext-real ) Real) ) : ( ( ) ( V30() real ext-real ) Element of REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) ,x0 : ( ( ) ( V30() real ext-real ) Real) .[ : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) & ( for
g being ( ( ) (
V30()
real ext-real )
Real) st
g : ( ( ) (
V30()
real ext-real )
Real)
in (dom f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) )
/\ ].(x0 : ( ( ) ( V30() real ext-real ) Real) - r : ( ( ) ( V30() real ext-real ) Real) ) : ( ( ) ( V30() real ext-real ) Element of REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) ,x0 : ( ( ) ( V30() real ext-real ) Real) .[ : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) holds
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
. g : ( ( ) (
V30()
real ext-real )
Real) : ( ( ) (
V30()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
<= f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
. g : ( ( ) (
V30()
real ext-real )
Real) : ( ( ) (
V30()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) ) ) holds
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
is_left_divergent_to+infty_in x0 : ( ( ) (
V30()
real ext-real )
Real) ;
theorem
for
x0 being ( ( ) (
V30()
real ext-real )
Real)
for
f1,
f being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,) st
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
is_left_divergent_to-infty_in x0 : ( ( ) (
V30()
real ext-real )
Real) & ( for
r being ( ( ) (
V30()
real ext-real )
Real) st
r : ( ( ) (
V30()
real ext-real )
Real)
< x0 : ( ( ) (
V30()
real ext-real )
Real) holds
ex
g being ( ( ) (
V30()
real ext-real )
Real) st
(
r : ( ( ) (
V30()
real ext-real )
Real)
< g : ( ( ) (
V30()
real ext-real )
Real) &
g : ( ( ) (
V30()
real ext-real )
Real)
< x0 : ( ( ) (
V30()
real ext-real )
Real) &
g : ( ( ) (
V30()
real ext-real )
Real)
in dom f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) ) ) & ex
r being ( ( ) (
V30()
real ext-real )
Real) st
(
0 : ( ( ) (
empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V30()
real ext-real non
positive non
negative V71()
V72()
V73()
V74()
V75()
V76()
V77()
V87()
V88() )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) )
< r : ( ( ) (
V30()
real ext-real )
Real) &
(dom f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) )
/\ ].(x0 : ( ( ) ( V30() real ext-real ) Real) - r : ( ( ) ( V30() real ext-real ) Real) ) : ( ( ) ( V30() real ext-real ) Element of REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) ,x0 : ( ( ) ( V30() real ext-real ) Real) .[ : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) )
c= (dom f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) )
/\ ].(x0 : ( ( ) ( V30() real ext-real ) Real) - r : ( ( ) ( V30() real ext-real ) Real) ) : ( ( ) ( V30() real ext-real ) Element of REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) ,x0 : ( ( ) ( V30() real ext-real ) Real) .[ : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) & ( for
g being ( ( ) (
V30()
real ext-real )
Real) st
g : ( ( ) (
V30()
real ext-real )
Real)
in (dom f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) )
/\ ].(x0 : ( ( ) ( V30() real ext-real ) Real) - r : ( ( ) ( V30() real ext-real ) Real) ) : ( ( ) ( V30() real ext-real ) Element of REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) ,x0 : ( ( ) ( V30() real ext-real ) Real) .[ : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) holds
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
. g : ( ( ) (
V30()
real ext-real )
Real) : ( ( ) (
V30()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
<= f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
. g : ( ( ) (
V30()
real ext-real )
Real) : ( ( ) (
V30()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) ) ) holds
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
is_left_divergent_to-infty_in x0 : ( ( ) (
V30()
real ext-real )
Real) ;
theorem
for
x0 being ( ( ) (
V30()
real ext-real )
Real)
for
f1,
f being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,) st
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
is_right_divergent_to+infty_in x0 : ( ( ) (
V30()
real ext-real )
Real) & ( for
r being ( ( ) (
V30()
real ext-real )
Real) st
x0 : ( ( ) (
V30()
real ext-real )
Real)
< r : ( ( ) (
V30()
real ext-real )
Real) holds
ex
g being ( ( ) (
V30()
real ext-real )
Real) st
(
g : ( ( ) (
V30()
real ext-real )
Real)
< r : ( ( ) (
V30()
real ext-real )
Real) &
x0 : ( ( ) (
V30()
real ext-real )
Real)
< g : ( ( ) (
V30()
real ext-real )
Real) &
g : ( ( ) (
V30()
real ext-real )
Real)
in dom f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) ) ) & ex
r being ( ( ) (
V30()
real ext-real )
Real) st
(
0 : ( ( ) (
empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V30()
real ext-real non
positive non
negative V71()
V72()
V73()
V74()
V75()
V76()
V77()
V87()
V88() )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) )
< r : ( ( ) (
V30()
real ext-real )
Real) &
(dom f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) )
/\ ].x0 : ( ( ) ( V30() real ext-real ) Real) ,(x0 : ( ( ) ( V30() real ext-real ) Real) + r : ( ( ) ( V30() real ext-real ) Real) ) : ( ( ) ( V30() real ext-real ) Element of REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) .[ : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) )
c= (dom f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) )
/\ ].x0 : ( ( ) ( V30() real ext-real ) Real) ,(x0 : ( ( ) ( V30() real ext-real ) Real) + r : ( ( ) ( V30() real ext-real ) Real) ) : ( ( ) ( V30() real ext-real ) Element of REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) .[ : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) & ( for
g being ( ( ) (
V30()
real ext-real )
Real) st
g : ( ( ) (
V30()
real ext-real )
Real)
in (dom f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) )
/\ ].x0 : ( ( ) ( V30() real ext-real ) Real) ,(x0 : ( ( ) ( V30() real ext-real ) Real) + r : ( ( ) ( V30() real ext-real ) Real) ) : ( ( ) ( V30() real ext-real ) Element of REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) .[ : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) holds
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
. g : ( ( ) (
V30()
real ext-real )
Real) : ( ( ) (
V30()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
<= f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
. g : ( ( ) (
V30()
real ext-real )
Real) : ( ( ) (
V30()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) ) ) holds
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
is_right_divergent_to+infty_in x0 : ( ( ) (
V30()
real ext-real )
Real) ;
theorem
for
x0 being ( ( ) (
V30()
real ext-real )
Real)
for
f1,
f being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,) st
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
is_right_divergent_to-infty_in x0 : ( ( ) (
V30()
real ext-real )
Real) & ( for
r being ( ( ) (
V30()
real ext-real )
Real) st
x0 : ( ( ) (
V30()
real ext-real )
Real)
< r : ( ( ) (
V30()
real ext-real )
Real) holds
ex
g being ( ( ) (
V30()
real ext-real )
Real) st
(
g : ( ( ) (
V30()
real ext-real )
Real)
< r : ( ( ) (
V30()
real ext-real )
Real) &
x0 : ( ( ) (
V30()
real ext-real )
Real)
< g : ( ( ) (
V30()
real ext-real )
Real) &
g : ( ( ) (
V30()
real ext-real )
Real)
in dom f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) ) ) & ex
r being ( ( ) (
V30()
real ext-real )
Real) st
(
0 : ( ( ) (
empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V30()
real ext-real non
positive non
negative V71()
V72()
V73()
V74()
V75()
V76()
V77()
V87()
V88() )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) )
< r : ( ( ) (
V30()
real ext-real )
Real) &
(dom f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) )
/\ ].x0 : ( ( ) ( V30() real ext-real ) Real) ,(x0 : ( ( ) ( V30() real ext-real ) Real) + r : ( ( ) ( V30() real ext-real ) Real) ) : ( ( ) ( V30() real ext-real ) Element of REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) .[ : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) )
c= (dom f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) )
/\ ].x0 : ( ( ) ( V30() real ext-real ) Real) ,(x0 : ( ( ) ( V30() real ext-real ) Real) + r : ( ( ) ( V30() real ext-real ) Real) ) : ( ( ) ( V30() real ext-real ) Element of REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) .[ : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) & ( for
g being ( ( ) (
V30()
real ext-real )
Real) st
g : ( ( ) (
V30()
real ext-real )
Real)
in (dom f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) )
/\ ].x0 : ( ( ) ( V30() real ext-real ) Real) ,(x0 : ( ( ) ( V30() real ext-real ) Real) + r : ( ( ) ( V30() real ext-real ) Real) ) : ( ( ) ( V30() real ext-real ) Element of REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) .[ : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) holds
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
. g : ( ( ) (
V30()
real ext-real )
Real) : ( ( ) (
V30()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
<= f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
. g : ( ( ) (
V30()
real ext-real )
Real) : ( ( ) (
V30()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) ) ) holds
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
is_right_divergent_to-infty_in x0 : ( ( ) (
V30()
real ext-real )
Real) ;
theorem
for
x0 being ( ( ) (
V30()
real ext-real )
Real)
for
f1,
f being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,) st
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
is_left_divergent_to+infty_in x0 : ( ( ) (
V30()
real ext-real )
Real) & ex
r being ( ( ) (
V30()
real ext-real )
Real) st
(
0 : ( ( ) (
empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V30()
real ext-real non
positive non
negative V71()
V72()
V73()
V74()
V75()
V76()
V77()
V87()
V88() )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) )
< r : ( ( ) (
V30()
real ext-real )
Real) &
].(x0 : ( ( ) ( V30() real ext-real ) Real) - r : ( ( ) ( V30() real ext-real ) Real) ) : ( ( ) ( V30() real ext-real ) Element of REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) ,x0 : ( ( ) ( V30() real ext-real ) Real) .[ : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) )
c= (dom f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) )
/\ (dom f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) & ( for
g being ( ( ) (
V30()
real ext-real )
Real) st
g : ( ( ) (
V30()
real ext-real )
Real)
in ].(x0 : ( ( ) ( V30() real ext-real ) Real) - r : ( ( ) ( V30() real ext-real ) Real) ) : ( ( ) ( V30() real ext-real ) Element of REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) ,x0 : ( ( ) ( V30() real ext-real ) Real) .[ : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) holds
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
. g : ( ( ) (
V30()
real ext-real )
Real) : ( ( ) (
V30()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
<= f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
. g : ( ( ) (
V30()
real ext-real )
Real) : ( ( ) (
V30()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) ) ) holds
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
is_left_divergent_to+infty_in x0 : ( ( ) (
V30()
real ext-real )
Real) ;
theorem
for
x0 being ( ( ) (
V30()
real ext-real )
Real)
for
f1,
f being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,) st
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
is_left_divergent_to-infty_in x0 : ( ( ) (
V30()
real ext-real )
Real) & ex
r being ( ( ) (
V30()
real ext-real )
Real) st
(
0 : ( ( ) (
empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V30()
real ext-real non
positive non
negative V71()
V72()
V73()
V74()
V75()
V76()
V77()
V87()
V88() )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) )
< r : ( ( ) (
V30()
real ext-real )
Real) &
].(x0 : ( ( ) ( V30() real ext-real ) Real) - r : ( ( ) ( V30() real ext-real ) Real) ) : ( ( ) ( V30() real ext-real ) Element of REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) ,x0 : ( ( ) ( V30() real ext-real ) Real) .[ : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) )
c= (dom f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) )
/\ (dom f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) & ( for
g being ( ( ) (
V30()
real ext-real )
Real) st
g : ( ( ) (
V30()
real ext-real )
Real)
in ].(x0 : ( ( ) ( V30() real ext-real ) Real) - r : ( ( ) ( V30() real ext-real ) Real) ) : ( ( ) ( V30() real ext-real ) Element of REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) ,x0 : ( ( ) ( V30() real ext-real ) Real) .[ : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) holds
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
. g : ( ( ) (
V30()
real ext-real )
Real) : ( ( ) (
V30()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
<= f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
. g : ( ( ) (
V30()
real ext-real )
Real) : ( ( ) (
V30()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) ) ) holds
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
is_left_divergent_to-infty_in x0 : ( ( ) (
V30()
real ext-real )
Real) ;
theorem
for
x0 being ( ( ) (
V30()
real ext-real )
Real)
for
f1,
f being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,) st
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
is_right_divergent_to+infty_in x0 : ( ( ) (
V30()
real ext-real )
Real) & ex
r being ( ( ) (
V30()
real ext-real )
Real) st
(
0 : ( ( ) (
empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V30()
real ext-real non
positive non
negative V71()
V72()
V73()
V74()
V75()
V76()
V77()
V87()
V88() )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) )
< r : ( ( ) (
V30()
real ext-real )
Real) &
].x0 : ( ( ) ( V30() real ext-real ) Real) ,(x0 : ( ( ) ( V30() real ext-real ) Real) + r : ( ( ) ( V30() real ext-real ) Real) ) : ( ( ) ( V30() real ext-real ) Element of REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) .[ : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) )
c= (dom f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) )
/\ (dom f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) & ( for
g being ( ( ) (
V30()
real ext-real )
Real) st
g : ( ( ) (
V30()
real ext-real )
Real)
in ].x0 : ( ( ) ( V30() real ext-real ) Real) ,(x0 : ( ( ) ( V30() real ext-real ) Real) + r : ( ( ) ( V30() real ext-real ) Real) ) : ( ( ) ( V30() real ext-real ) Element of REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) .[ : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) holds
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
. g : ( ( ) (
V30()
real ext-real )
Real) : ( ( ) (
V30()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
<= f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
. g : ( ( ) (
V30()
real ext-real )
Real) : ( ( ) (
V30()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) ) ) holds
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
is_right_divergent_to+infty_in x0 : ( ( ) (
V30()
real ext-real )
Real) ;
theorem
for
x0 being ( ( ) (
V30()
real ext-real )
Real)
for
f1,
f being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,) st
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
is_right_divergent_to-infty_in x0 : ( ( ) (
V30()
real ext-real )
Real) & ex
r being ( ( ) (
V30()
real ext-real )
Real) st
(
0 : ( ( ) (
empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V30()
real ext-real non
positive non
negative V71()
V72()
V73()
V74()
V75()
V76()
V77()
V87()
V88() )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) )
< r : ( ( ) (
V30()
real ext-real )
Real) &
].x0 : ( ( ) ( V30() real ext-real ) Real) ,(x0 : ( ( ) ( V30() real ext-real ) Real) + r : ( ( ) ( V30() real ext-real ) Real) ) : ( ( ) ( V30() real ext-real ) Element of REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) .[ : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) )
c= (dom f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) )
/\ (dom f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) & ( for
g being ( ( ) (
V30()
real ext-real )
Real) st
g : ( ( ) (
V30()
real ext-real )
Real)
in ].x0 : ( ( ) ( V30() real ext-real ) Real) ,(x0 : ( ( ) ( V30() real ext-real ) Real) + r : ( ( ) ( V30() real ext-real ) Real) ) : ( ( ) ( V30() real ext-real ) Element of REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) .[ : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) holds
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
. g : ( ( ) (
V30()
real ext-real )
Real) : ( ( ) (
V30()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
<= f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
. g : ( ( ) (
V30()
real ext-real )
Real) : ( ( ) (
V30()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) ) ) holds
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
is_right_divergent_to-infty_in x0 : ( ( ) (
V30()
real ext-real )
Real) ;
theorem
for
x0,
r being ( ( ) (
V30()
real ext-real )
Real)
for
f being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,) st
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
is_left_convergent_in x0 : ( ( ) (
V30()
real ext-real )
Real) holds
(
r : ( ( ) (
V30()
real ext-real )
Real)
(#) f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
is_left_convergent_in x0 : ( ( ) (
V30()
real ext-real )
Real) &
lim_left (
(r : ( ( ) ( V30() real ext-real ) Real) (#) f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) ,
x0 : ( ( ) (
V30()
real ext-real )
Real) ) : ( ( ) (
V30()
real ext-real )
Real)
= r : ( ( ) (
V30()
real ext-real )
Real)
* (lim_left (f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,x0 : ( ( ) ( V30() real ext-real ) Real) )) : ( ( ) (
V30()
real ext-real )
Real) : ( ( ) (
V30()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) ) ;
theorem
for
x0 being ( ( ) (
V30()
real ext-real )
Real)
for
f1,
f2 being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,) st
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
is_left_convergent_in x0 : ( ( ) (
V30()
real ext-real )
Real) &
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
is_left_convergent_in x0 : ( ( ) (
V30()
real ext-real )
Real) & ( for
r being ( ( ) (
V30()
real ext-real )
Real) st
r : ( ( ) (
V30()
real ext-real )
Real)
< x0 : ( ( ) (
V30()
real ext-real )
Real) holds
ex
g being ( ( ) (
V30()
real ext-real )
Real) st
(
r : ( ( ) (
V30()
real ext-real )
Real)
< g : ( ( ) (
V30()
real ext-real )
Real) &
g : ( ( ) (
V30()
real ext-real )
Real)
< x0 : ( ( ) (
V30()
real ext-real )
Real) &
g : ( ( ) (
V30()
real ext-real )
Real)
in dom (f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) + f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) ) ) holds
(
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
+ f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
is_left_convergent_in x0 : ( ( ) (
V30()
real ext-real )
Real) &
lim_left (
(f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) + f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) ,
x0 : ( ( ) (
V30()
real ext-real )
Real) ) : ( ( ) (
V30()
real ext-real )
Real)
= (lim_left (f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,x0 : ( ( ) ( V30() real ext-real ) Real) )) : ( ( ) (
V30()
real ext-real )
Real)
+ (lim_left (f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,x0 : ( ( ) ( V30() real ext-real ) Real) )) : ( ( ) (
V30()
real ext-real )
Real) : ( ( ) (
V30()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) ) ;
theorem
for
x0 being ( ( ) (
V30()
real ext-real )
Real)
for
f1,
f2 being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,) st
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
is_left_convergent_in x0 : ( ( ) (
V30()
real ext-real )
Real) &
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
is_left_convergent_in x0 : ( ( ) (
V30()
real ext-real )
Real) & ( for
r being ( ( ) (
V30()
real ext-real )
Real) st
r : ( ( ) (
V30()
real ext-real )
Real)
< x0 : ( ( ) (
V30()
real ext-real )
Real) holds
ex
g being ( ( ) (
V30()
real ext-real )
Real) st
(
r : ( ( ) (
V30()
real ext-real )
Real)
< g : ( ( ) (
V30()
real ext-real )
Real) &
g : ( ( ) (
V30()
real ext-real )
Real)
< x0 : ( ( ) (
V30()
real ext-real )
Real) &
g : ( ( ) (
V30()
real ext-real )
Real)
in dom (f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) - f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) ) ) holds
(
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
- f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
is_left_convergent_in x0 : ( ( ) (
V30()
real ext-real )
Real) &
lim_left (
(f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) - f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) ,
x0 : ( ( ) (
V30()
real ext-real )
Real) ) : ( ( ) (
V30()
real ext-real )
Real)
= (lim_left (f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,x0 : ( ( ) ( V30() real ext-real ) Real) )) : ( ( ) (
V30()
real ext-real )
Real)
- (lim_left (f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,x0 : ( ( ) ( V30() real ext-real ) Real) )) : ( ( ) (
V30()
real ext-real )
Real) : ( ( ) (
V30()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) ) ;
theorem
for
x0 being ( ( ) (
V30()
real ext-real )
Real)
for
f being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,) st
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
is_left_convergent_in x0 : ( ( ) (
V30()
real ext-real )
Real) &
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
" {0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V30() real ext-real non positive non negative V71() V72() V73() V74() V75() V76() V77() V87() V88() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() ) Element of K19(REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) : ( ( ) ( ) set ) ) ) } : ( ( ) ( non
empty V71()
V72()
V73()
V74()
V75()
V76() )
set ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) )
= {} : ( ( ) (
empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V30()
real ext-real non
positive non
negative V71()
V72()
V73()
V74()
V75()
V76()
V77() )
set ) &
lim_left (
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,) ,
x0 : ( ( ) (
V30()
real ext-real )
Real) ) : ( ( ) (
V30()
real ext-real )
Real)
<> 0 : ( ( ) (
empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V30()
real ext-real non
positive non
negative V71()
V72()
V73()
V74()
V75()
V76()
V77()
V87()
V88() )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) ) holds
(
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
^ : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
is_left_convergent_in x0 : ( ( ) (
V30()
real ext-real )
Real) &
lim_left (
(f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) ^) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) ,
x0 : ( ( ) (
V30()
real ext-real )
Real) ) : ( ( ) (
V30()
real ext-real )
Real)
= (lim_left (f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,x0 : ( ( ) ( V30() real ext-real ) Real) )) : ( ( ) (
V30()
real ext-real )
Real)
" : ( ( ) (
V30()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) ) ;
theorem
for
x0 being ( ( ) (
V30()
real ext-real )
Real)
for
f being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,) st
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
is_left_convergent_in x0 : ( ( ) (
V30()
real ext-real )
Real) &
lim_left (
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,) ,
x0 : ( ( ) (
V30()
real ext-real )
Real) ) : ( ( ) (
V30()
real ext-real )
Real)
<> 0 : ( ( ) (
empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V30()
real ext-real non
positive non
negative V71()
V72()
V73()
V74()
V75()
V76()
V77()
V87()
V88() )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) ) & ( for
r being ( ( ) (
V30()
real ext-real )
Real) st
r : ( ( ) (
V30()
real ext-real )
Real)
< x0 : ( ( ) (
V30()
real ext-real )
Real) holds
ex
g being ( ( ) (
V30()
real ext-real )
Real) st
(
r : ( ( ) (
V30()
real ext-real )
Real)
< g : ( ( ) (
V30()
real ext-real )
Real) &
g : ( ( ) (
V30()
real ext-real )
Real)
< x0 : ( ( ) (
V30()
real ext-real )
Real) &
g : ( ( ) (
V30()
real ext-real )
Real)
in dom f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) &
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
. g : ( ( ) (
V30()
real ext-real )
Real) : ( ( ) (
V30()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
<> 0 : ( ( ) (
empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V30()
real ext-real non
positive non
negative V71()
V72()
V73()
V74()
V75()
V76()
V77()
V87()
V88() )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) ) ) ) holds
(
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
^ : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
is_left_convergent_in x0 : ( ( ) (
V30()
real ext-real )
Real) &
lim_left (
(f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) ^) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) ,
x0 : ( ( ) (
V30()
real ext-real )
Real) ) : ( ( ) (
V30()
real ext-real )
Real)
= (lim_left (f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,x0 : ( ( ) ( V30() real ext-real ) Real) )) : ( ( ) (
V30()
real ext-real )
Real)
" : ( ( ) (
V30()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) ) ;
theorem
for
x0 being ( ( ) (
V30()
real ext-real )
Real)
for
f1,
f2 being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,) st
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
is_left_convergent_in x0 : ( ( ) (
V30()
real ext-real )
Real) &
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
is_left_convergent_in x0 : ( ( ) (
V30()
real ext-real )
Real) & ( for
r being ( ( ) (
V30()
real ext-real )
Real) st
r : ( ( ) (
V30()
real ext-real )
Real)
< x0 : ( ( ) (
V30()
real ext-real )
Real) holds
ex
g being ( ( ) (
V30()
real ext-real )
Real) st
(
r : ( ( ) (
V30()
real ext-real )
Real)
< g : ( ( ) (
V30()
real ext-real )
Real) &
g : ( ( ) (
V30()
real ext-real )
Real)
< x0 : ( ( ) (
V30()
real ext-real )
Real) &
g : ( ( ) (
V30()
real ext-real )
Real)
in dom (f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) (#) f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) ) ) holds
(
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
(#) f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
is_left_convergent_in x0 : ( ( ) (
V30()
real ext-real )
Real) &
lim_left (
(f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) (#) f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) ,
x0 : ( ( ) (
V30()
real ext-real )
Real) ) : ( ( ) (
V30()
real ext-real )
Real)
= (lim_left (f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,x0 : ( ( ) ( V30() real ext-real ) Real) )) : ( ( ) (
V30()
real ext-real )
Real)
* (lim_left (f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,x0 : ( ( ) ( V30() real ext-real ) Real) )) : ( ( ) (
V30()
real ext-real )
Real) : ( ( ) (
V30()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) ) ;
theorem
for
x0 being ( ( ) (
V30()
real ext-real )
Real)
for
f1,
f2 being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,) st
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
is_left_convergent_in x0 : ( ( ) (
V30()
real ext-real )
Real) &
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
is_left_convergent_in x0 : ( ( ) (
V30()
real ext-real )
Real) &
lim_left (
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,) ,
x0 : ( ( ) (
V30()
real ext-real )
Real) ) : ( ( ) (
V30()
real ext-real )
Real)
<> 0 : ( ( ) (
empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V30()
real ext-real non
positive non
negative V71()
V72()
V73()
V74()
V75()
V76()
V77()
V87()
V88() )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) ) & ( for
r being ( ( ) (
V30()
real ext-real )
Real) st
r : ( ( ) (
V30()
real ext-real )
Real)
< x0 : ( ( ) (
V30()
real ext-real )
Real) holds
ex
g being ( ( ) (
V30()
real ext-real )
Real) st
(
r : ( ( ) (
V30()
real ext-real )
Real)
< g : ( ( ) (
V30()
real ext-real )
Real) &
g : ( ( ) (
V30()
real ext-real )
Real)
< x0 : ( ( ) (
V30()
real ext-real )
Real) &
g : ( ( ) (
V30()
real ext-real )
Real)
in dom (f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) / f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) ) ) holds
(
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
/ f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
is_left_convergent_in x0 : ( ( ) (
V30()
real ext-real )
Real) &
lim_left (
(f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) / f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) ,
x0 : ( ( ) (
V30()
real ext-real )
Real) ) : ( ( ) (
V30()
real ext-real )
Real)
= (lim_left (f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,x0 : ( ( ) ( V30() real ext-real ) Real) )) : ( ( ) (
V30()
real ext-real )
Real)
/ (lim_left (f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,x0 : ( ( ) ( V30() real ext-real ) Real) )) : ( ( ) (
V30()
real ext-real )
Real) : ( ( ) (
V30()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) ) ;
theorem
for
x0,
r being ( ( ) (
V30()
real ext-real )
Real)
for
f being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,) st
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
is_right_convergent_in x0 : ( ( ) (
V30()
real ext-real )
Real) holds
(
r : ( ( ) (
V30()
real ext-real )
Real)
(#) f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
is_right_convergent_in x0 : ( ( ) (
V30()
real ext-real )
Real) &
lim_right (
(r : ( ( ) ( V30() real ext-real ) Real) (#) f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) ,
x0 : ( ( ) (
V30()
real ext-real )
Real) ) : ( ( ) (
V30()
real ext-real )
Real)
= r : ( ( ) (
V30()
real ext-real )
Real)
* (lim_right (f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,x0 : ( ( ) ( V30() real ext-real ) Real) )) : ( ( ) (
V30()
real ext-real )
Real) : ( ( ) (
V30()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) ) ;
theorem
for
x0 being ( ( ) (
V30()
real ext-real )
Real)
for
f1,
f2 being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,) st
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
is_right_convergent_in x0 : ( ( ) (
V30()
real ext-real )
Real) &
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
is_right_convergent_in x0 : ( ( ) (
V30()
real ext-real )
Real) & ( for
r being ( ( ) (
V30()
real ext-real )
Real) st
x0 : ( ( ) (
V30()
real ext-real )
Real)
< r : ( ( ) (
V30()
real ext-real )
Real) holds
ex
g being ( ( ) (
V30()
real ext-real )
Real) st
(
g : ( ( ) (
V30()
real ext-real )
Real)
< r : ( ( ) (
V30()
real ext-real )
Real) &
x0 : ( ( ) (
V30()
real ext-real )
Real)
< g : ( ( ) (
V30()
real ext-real )
Real) &
g : ( ( ) (
V30()
real ext-real )
Real)
in dom (f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) + f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) ) ) holds
(
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
+ f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
is_right_convergent_in x0 : ( ( ) (
V30()
real ext-real )
Real) &
lim_right (
(f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) + f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) ,
x0 : ( ( ) (
V30()
real ext-real )
Real) ) : ( ( ) (
V30()
real ext-real )
Real)
= (lim_right (f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,x0 : ( ( ) ( V30() real ext-real ) Real) )) : ( ( ) (
V30()
real ext-real )
Real)
+ (lim_right (f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,x0 : ( ( ) ( V30() real ext-real ) Real) )) : ( ( ) (
V30()
real ext-real )
Real) : ( ( ) (
V30()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) ) ;
theorem
for
x0 being ( ( ) (
V30()
real ext-real )
Real)
for
f1,
f2 being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,) st
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
is_right_convergent_in x0 : ( ( ) (
V30()
real ext-real )
Real) &
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
is_right_convergent_in x0 : ( ( ) (
V30()
real ext-real )
Real) & ( for
r being ( ( ) (
V30()
real ext-real )
Real) st
x0 : ( ( ) (
V30()
real ext-real )
Real)
< r : ( ( ) (
V30()
real ext-real )
Real) holds
ex
g being ( ( ) (
V30()
real ext-real )
Real) st
(
g : ( ( ) (
V30()
real ext-real )
Real)
< r : ( ( ) (
V30()
real ext-real )
Real) &
x0 : ( ( ) (
V30()
real ext-real )
Real)
< g : ( ( ) (
V30()
real ext-real )
Real) &
g : ( ( ) (
V30()
real ext-real )
Real)
in dom (f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) - f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) ) ) holds
(
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
- f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
is_right_convergent_in x0 : ( ( ) (
V30()
real ext-real )
Real) &
lim_right (
(f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) - f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) ,
x0 : ( ( ) (
V30()
real ext-real )
Real) ) : ( ( ) (
V30()
real ext-real )
Real)
= (lim_right (f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,x0 : ( ( ) ( V30() real ext-real ) Real) )) : ( ( ) (
V30()
real ext-real )
Real)
- (lim_right (f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,x0 : ( ( ) ( V30() real ext-real ) Real) )) : ( ( ) (
V30()
real ext-real )
Real) : ( ( ) (
V30()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) ) ;
theorem
for
x0 being ( ( ) (
V30()
real ext-real )
Real)
for
f being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,) st
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
is_right_convergent_in x0 : ( ( ) (
V30()
real ext-real )
Real) &
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
" {0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V30() real ext-real non positive non negative V71() V72() V73() V74() V75() V76() V77() V87() V88() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() ) Element of K19(REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) : ( ( ) ( ) set ) ) ) } : ( ( ) ( non
empty V71()
V72()
V73()
V74()
V75()
V76() )
set ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) )
= {} : ( ( ) (
empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V30()
real ext-real non
positive non
negative V71()
V72()
V73()
V74()
V75()
V76()
V77() )
set ) &
lim_right (
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,) ,
x0 : ( ( ) (
V30()
real ext-real )
Real) ) : ( ( ) (
V30()
real ext-real )
Real)
<> 0 : ( ( ) (
empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V30()
real ext-real non
positive non
negative V71()
V72()
V73()
V74()
V75()
V76()
V77()
V87()
V88() )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) ) holds
(
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
^ : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
is_right_convergent_in x0 : ( ( ) (
V30()
real ext-real )
Real) &
lim_right (
(f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) ^) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) ,
x0 : ( ( ) (
V30()
real ext-real )
Real) ) : ( ( ) (
V30()
real ext-real )
Real)
= (lim_right (f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,x0 : ( ( ) ( V30() real ext-real ) Real) )) : ( ( ) (
V30()
real ext-real )
Real)
" : ( ( ) (
V30()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) ) ;
theorem
for
x0 being ( ( ) (
V30()
real ext-real )
Real)
for
f being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,) st
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
is_right_convergent_in x0 : ( ( ) (
V30()
real ext-real )
Real) &
lim_right (
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,) ,
x0 : ( ( ) (
V30()
real ext-real )
Real) ) : ( ( ) (
V30()
real ext-real )
Real)
<> 0 : ( ( ) (
empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V30()
real ext-real non
positive non
negative V71()
V72()
V73()
V74()
V75()
V76()
V77()
V87()
V88() )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) ) & ( for
r being ( ( ) (
V30()
real ext-real )
Real) st
x0 : ( ( ) (
V30()
real ext-real )
Real)
< r : ( ( ) (
V30()
real ext-real )
Real) holds
ex
g being ( ( ) (
V30()
real ext-real )
Real) st
(
g : ( ( ) (
V30()
real ext-real )
Real)
< r : ( ( ) (
V30()
real ext-real )
Real) &
x0 : ( ( ) (
V30()
real ext-real )
Real)
< g : ( ( ) (
V30()
real ext-real )
Real) &
g : ( ( ) (
V30()
real ext-real )
Real)
in dom f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) &
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
. g : ( ( ) (
V30()
real ext-real )
Real) : ( ( ) (
V30()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
<> 0 : ( ( ) (
empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V30()
real ext-real non
positive non
negative V71()
V72()
V73()
V74()
V75()
V76()
V77()
V87()
V88() )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) ) ) ) holds
(
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
^ : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
is_right_convergent_in x0 : ( ( ) (
V30()
real ext-real )
Real) &
lim_right (
(f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) ^) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) ,
x0 : ( ( ) (
V30()
real ext-real )
Real) ) : ( ( ) (
V30()
real ext-real )
Real)
= (lim_right (f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,x0 : ( ( ) ( V30() real ext-real ) Real) )) : ( ( ) (
V30()
real ext-real )
Real)
" : ( ( ) (
V30()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) ) ;
theorem
for
x0 being ( ( ) (
V30()
real ext-real )
Real)
for
f1,
f2 being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,) st
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
is_right_convergent_in x0 : ( ( ) (
V30()
real ext-real )
Real) &
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
is_right_convergent_in x0 : ( ( ) (
V30()
real ext-real )
Real) & ( for
r being ( ( ) (
V30()
real ext-real )
Real) st
x0 : ( ( ) (
V30()
real ext-real )
Real)
< r : ( ( ) (
V30()
real ext-real )
Real) holds
ex
g being ( ( ) (
V30()
real ext-real )
Real) st
(
g : ( ( ) (
V30()
real ext-real )
Real)
< r : ( ( ) (
V30()
real ext-real )
Real) &
x0 : ( ( ) (
V30()
real ext-real )
Real)
< g : ( ( ) (
V30()
real ext-real )
Real) &
g : ( ( ) (
V30()
real ext-real )
Real)
in dom (f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) (#) f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) ) ) holds
(
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
(#) f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
is_right_convergent_in x0 : ( ( ) (
V30()
real ext-real )
Real) &
lim_right (
(f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) (#) f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) ,
x0 : ( ( ) (
V30()
real ext-real )
Real) ) : ( ( ) (
V30()
real ext-real )
Real)
= (lim_right (f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,x0 : ( ( ) ( V30() real ext-real ) Real) )) : ( ( ) (
V30()
real ext-real )
Real)
* (lim_right (f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,x0 : ( ( ) ( V30() real ext-real ) Real) )) : ( ( ) (
V30()
real ext-real )
Real) : ( ( ) (
V30()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) ) ;
theorem
for
x0 being ( ( ) (
V30()
real ext-real )
Real)
for
f1,
f2 being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,) st
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
is_right_convergent_in x0 : ( ( ) (
V30()
real ext-real )
Real) &
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
is_right_convergent_in x0 : ( ( ) (
V30()
real ext-real )
Real) &
lim_right (
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,) ,
x0 : ( ( ) (
V30()
real ext-real )
Real) ) : ( ( ) (
V30()
real ext-real )
Real)
<> 0 : ( ( ) (
empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V30()
real ext-real non
positive non
negative V71()
V72()
V73()
V74()
V75()
V76()
V77()
V87()
V88() )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) ) & ( for
r being ( ( ) (
V30()
real ext-real )
Real) st
x0 : ( ( ) (
V30()
real ext-real )
Real)
< r : ( ( ) (
V30()
real ext-real )
Real) holds
ex
g being ( ( ) (
V30()
real ext-real )
Real) st
(
g : ( ( ) (
V30()
real ext-real )
Real)
< r : ( ( ) (
V30()
real ext-real )
Real) &
x0 : ( ( ) (
V30()
real ext-real )
Real)
< g : ( ( ) (
V30()
real ext-real )
Real) &
g : ( ( ) (
V30()
real ext-real )
Real)
in dom (f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) / f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) ) ) holds
(
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
/ f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
is_right_convergent_in x0 : ( ( ) (
V30()
real ext-real )
Real) &
lim_right (
(f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) / f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) ,
x0 : ( ( ) (
V30()
real ext-real )
Real) ) : ( ( ) (
V30()
real ext-real )
Real)
= (lim_right (f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,x0 : ( ( ) ( V30() real ext-real ) Real) )) : ( ( ) (
V30()
real ext-real )
Real)
/ (lim_right (f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,x0 : ( ( ) ( V30() real ext-real ) Real) )) : ( ( ) (
V30()
real ext-real )
Real) : ( ( ) (
V30()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) ) ;
theorem
for
x0 being ( ( ) (
V30()
real ext-real )
Real)
for
f1,
f2 being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,) st
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
is_left_convergent_in x0 : ( ( ) (
V30()
real ext-real )
Real) &
lim_left (
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,) ,
x0 : ( ( ) (
V30()
real ext-real )
Real) ) : ( ( ) (
V30()
real ext-real )
Real)
= 0 : ( ( ) (
empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V30()
real ext-real non
positive non
negative V71()
V72()
V73()
V74()
V75()
V76()
V77()
V87()
V88() )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) ) & ( for
r being ( ( ) (
V30()
real ext-real )
Real) st
r : ( ( ) (
V30()
real ext-real )
Real)
< x0 : ( ( ) (
V30()
real ext-real )
Real) holds
ex
g being ( ( ) (
V30()
real ext-real )
Real) st
(
r : ( ( ) (
V30()
real ext-real )
Real)
< g : ( ( ) (
V30()
real ext-real )
Real) &
g : ( ( ) (
V30()
real ext-real )
Real)
< x0 : ( ( ) (
V30()
real ext-real )
Real) &
g : ( ( ) (
V30()
real ext-real )
Real)
in dom (f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) (#) f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) ) ) & ex
r being ( ( ) (
V30()
real ext-real )
Real) st
(
0 : ( ( ) (
empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V30()
real ext-real non
positive non
negative V71()
V72()
V73()
V74()
V75()
V76()
V77()
V87()
V88() )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) )
< r : ( ( ) (
V30()
real ext-real )
Real) &
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
| ].(x0 : ( ( ) ( V30() real ext-real ) Real) - r : ( ( ) ( V30() real ext-real ) Real) ) : ( ( ) ( V30() real ext-real ) Element of REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) ,x0 : ( ( ) ( V30() real ext-real ) Real) .[ : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) is
bounded ) holds
(
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
(#) f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
is_left_convergent_in x0 : ( ( ) (
V30()
real ext-real )
Real) &
lim_left (
(f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) (#) f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) ,
x0 : ( ( ) (
V30()
real ext-real )
Real) ) : ( ( ) (
V30()
real ext-real )
Real)
= 0 : ( ( ) (
empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V30()
real ext-real non
positive non
negative V71()
V72()
V73()
V74()
V75()
V76()
V77()
V87()
V88() )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) ) ) ;
theorem
for
x0 being ( ( ) (
V30()
real ext-real )
Real)
for
f1,
f2 being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,) st
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
is_right_convergent_in x0 : ( ( ) (
V30()
real ext-real )
Real) &
lim_right (
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,) ,
x0 : ( ( ) (
V30()
real ext-real )
Real) ) : ( ( ) (
V30()
real ext-real )
Real)
= 0 : ( ( ) (
empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V30()
real ext-real non
positive non
negative V71()
V72()
V73()
V74()
V75()
V76()
V77()
V87()
V88() )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) ) & ( for
r being ( ( ) (
V30()
real ext-real )
Real) st
x0 : ( ( ) (
V30()
real ext-real )
Real)
< r : ( ( ) (
V30()
real ext-real )
Real) holds
ex
g being ( ( ) (
V30()
real ext-real )
Real) st
(
g : ( ( ) (
V30()
real ext-real )
Real)
< r : ( ( ) (
V30()
real ext-real )
Real) &
x0 : ( ( ) (
V30()
real ext-real )
Real)
< g : ( ( ) (
V30()
real ext-real )
Real) &
g : ( ( ) (
V30()
real ext-real )
Real)
in dom (f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) (#) f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) ) ) & ex
r being ( ( ) (
V30()
real ext-real )
Real) st
(
0 : ( ( ) (
empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V30()
real ext-real non
positive non
negative V71()
V72()
V73()
V74()
V75()
V76()
V77()
V87()
V88() )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) )
< r : ( ( ) (
V30()
real ext-real )
Real) &
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
| ].x0 : ( ( ) ( V30() real ext-real ) Real) ,(x0 : ( ( ) ( V30() real ext-real ) Real) + r : ( ( ) ( V30() real ext-real ) Real) ) : ( ( ) ( V30() real ext-real ) Element of REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) .[ : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) is
bounded ) holds
(
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
(#) f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
is_right_convergent_in x0 : ( ( ) (
V30()
real ext-real )
Real) &
lim_right (
(f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) (#) f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) ,
x0 : ( ( ) (
V30()
real ext-real )
Real) ) : ( ( ) (
V30()
real ext-real )
Real)
= 0 : ( ( ) (
empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V30()
real ext-real non
positive non
negative V71()
V72()
V73()
V74()
V75()
V76()
V77()
V87()
V88() )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) ) ) ;
theorem
for
x0 being ( ( ) (
V30()
real ext-real )
Real)
for
f1,
f2,
f being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,) st
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
is_left_convergent_in x0 : ( ( ) (
V30()
real ext-real )
Real) &
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
is_left_convergent_in x0 : ( ( ) (
V30()
real ext-real )
Real) &
lim_left (
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,) ,
x0 : ( ( ) (
V30()
real ext-real )
Real) ) : ( ( ) (
V30()
real ext-real )
Real)
= lim_left (
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,) ,
x0 : ( ( ) (
V30()
real ext-real )
Real) ) : ( ( ) (
V30()
real ext-real )
Real) & ( for
r being ( ( ) (
V30()
real ext-real )
Real) st
r : ( ( ) (
V30()
real ext-real )
Real)
< x0 : ( ( ) (
V30()
real ext-real )
Real) holds
ex
g being ( ( ) (
V30()
real ext-real )
Real) st
(
r : ( ( ) (
V30()
real ext-real )
Real)
< g : ( ( ) (
V30()
real ext-real )
Real) &
g : ( ( ) (
V30()
real ext-real )
Real)
< x0 : ( ( ) (
V30()
real ext-real )
Real) &
g : ( ( ) (
V30()
real ext-real )
Real)
in dom f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) ) ) & ex
r being ( ( ) (
V30()
real ext-real )
Real) st
(
0 : ( ( ) (
empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V30()
real ext-real non
positive non
negative V71()
V72()
V73()
V74()
V75()
V76()
V77()
V87()
V88() )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) )
< r : ( ( ) (
V30()
real ext-real )
Real) & ( for
g being ( ( ) (
V30()
real ext-real )
Real) st
g : ( ( ) (
V30()
real ext-real )
Real)
in (dom f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) )
/\ ].(x0 : ( ( ) ( V30() real ext-real ) Real) - r : ( ( ) ( V30() real ext-real ) Real) ) : ( ( ) ( V30() real ext-real ) Element of REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) ,x0 : ( ( ) ( V30() real ext-real ) Real) .[ : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) holds
(
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
. g : ( ( ) (
V30()
real ext-real )
Real) : ( ( ) (
V30()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
<= f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
. g : ( ( ) (
V30()
real ext-real )
Real) : ( ( ) (
V30()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) &
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
. g : ( ( ) (
V30()
real ext-real )
Real) : ( ( ) (
V30()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
<= f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
. g : ( ( ) (
V30()
real ext-real )
Real) : ( ( ) (
V30()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) ) ) & ( (
(dom f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) )
/\ ].(x0 : ( ( ) ( V30() real ext-real ) Real) - r : ( ( ) ( V30() real ext-real ) Real) ) : ( ( ) ( V30() real ext-real ) Element of REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) ,x0 : ( ( ) ( V30() real ext-real ) Real) .[ : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) )
c= (dom f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) )
/\ ].(x0 : ( ( ) ( V30() real ext-real ) Real) - r : ( ( ) ( V30() real ext-real ) Real) ) : ( ( ) ( V30() real ext-real ) Element of REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) ,x0 : ( ( ) ( V30() real ext-real ) Real) .[ : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) &
(dom f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) )
/\ ].(x0 : ( ( ) ( V30() real ext-real ) Real) - r : ( ( ) ( V30() real ext-real ) Real) ) : ( ( ) ( V30() real ext-real ) Element of REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) ,x0 : ( ( ) ( V30() real ext-real ) Real) .[ : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) )
c= (dom f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) )
/\ ].(x0 : ( ( ) ( V30() real ext-real ) Real) - r : ( ( ) ( V30() real ext-real ) Real) ) : ( ( ) ( V30() real ext-real ) Element of REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) ,x0 : ( ( ) ( V30() real ext-real ) Real) .[ : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) ) or (
(dom f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) )
/\ ].(x0 : ( ( ) ( V30() real ext-real ) Real) - r : ( ( ) ( V30() real ext-real ) Real) ) : ( ( ) ( V30() real ext-real ) Element of REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) ,x0 : ( ( ) ( V30() real ext-real ) Real) .[ : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) )
c= (dom f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) )
/\ ].(x0 : ( ( ) ( V30() real ext-real ) Real) - r : ( ( ) ( V30() real ext-real ) Real) ) : ( ( ) ( V30() real ext-real ) Element of REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) ,x0 : ( ( ) ( V30() real ext-real ) Real) .[ : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) &
(dom f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) )
/\ ].(x0 : ( ( ) ( V30() real ext-real ) Real) - r : ( ( ) ( V30() real ext-real ) Real) ) : ( ( ) ( V30() real ext-real ) Element of REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) ,x0 : ( ( ) ( V30() real ext-real ) Real) .[ : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) )
c= (dom f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) )
/\ ].(x0 : ( ( ) ( V30() real ext-real ) Real) - r : ( ( ) ( V30() real ext-real ) Real) ) : ( ( ) ( V30() real ext-real ) Element of REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) ,x0 : ( ( ) ( V30() real ext-real ) Real) .[ : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) ) ) ) holds
(
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
is_left_convergent_in x0 : ( ( ) (
V30()
real ext-real )
Real) &
lim_left (
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,) ,
x0 : ( ( ) (
V30()
real ext-real )
Real) ) : ( ( ) (
V30()
real ext-real )
Real)
= lim_left (
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,) ,
x0 : ( ( ) (
V30()
real ext-real )
Real) ) : ( ( ) (
V30()
real ext-real )
Real) ) ;
theorem
for
x0 being ( ( ) (
V30()
real ext-real )
Real)
for
f1,
f2,
f being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,) st
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
is_left_convergent_in x0 : ( ( ) (
V30()
real ext-real )
Real) &
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
is_left_convergent_in x0 : ( ( ) (
V30()
real ext-real )
Real) &
lim_left (
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,) ,
x0 : ( ( ) (
V30()
real ext-real )
Real) ) : ( ( ) (
V30()
real ext-real )
Real)
= lim_left (
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,) ,
x0 : ( ( ) (
V30()
real ext-real )
Real) ) : ( ( ) (
V30()
real ext-real )
Real) & ex
r being ( ( ) (
V30()
real ext-real )
Real) st
(
0 : ( ( ) (
empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V30()
real ext-real non
positive non
negative V71()
V72()
V73()
V74()
V75()
V76()
V77()
V87()
V88() )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) )
< r : ( ( ) (
V30()
real ext-real )
Real) &
].(x0 : ( ( ) ( V30() real ext-real ) Real) - r : ( ( ) ( V30() real ext-real ) Real) ) : ( ( ) ( V30() real ext-real ) Element of REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) ,x0 : ( ( ) ( V30() real ext-real ) Real) .[ : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) )
c= ((dom f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( ( ) ( V71() V72() V73() ) Element of K19(REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) : ( ( ) ( ) set ) ) /\ (dom f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( ( ) ( V71() V72() V73() ) Element of K19(REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) )
/\ (dom f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) & ( for
g being ( ( ) (
V30()
real ext-real )
Real) st
g : ( ( ) (
V30()
real ext-real )
Real)
in ].(x0 : ( ( ) ( V30() real ext-real ) Real) - r : ( ( ) ( V30() real ext-real ) Real) ) : ( ( ) ( V30() real ext-real ) Element of REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) ,x0 : ( ( ) ( V30() real ext-real ) Real) .[ : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) holds
(
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
. g : ( ( ) (
V30()
real ext-real )
Real) : ( ( ) (
V30()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
<= f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
. g : ( ( ) (
V30()
real ext-real )
Real) : ( ( ) (
V30()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) &
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
. g : ( ( ) (
V30()
real ext-real )
Real) : ( ( ) (
V30()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
<= f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
. g : ( ( ) (
V30()
real ext-real )
Real) : ( ( ) (
V30()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) ) ) ) holds
(
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
is_left_convergent_in x0 : ( ( ) (
V30()
real ext-real )
Real) &
lim_left (
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,) ,
x0 : ( ( ) (
V30()
real ext-real )
Real) ) : ( ( ) (
V30()
real ext-real )
Real)
= lim_left (
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,) ,
x0 : ( ( ) (
V30()
real ext-real )
Real) ) : ( ( ) (
V30()
real ext-real )
Real) ) ;
theorem
for
x0 being ( ( ) (
V30()
real ext-real )
Real)
for
f1,
f2,
f being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,) st
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
is_right_convergent_in x0 : ( ( ) (
V30()
real ext-real )
Real) &
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
is_right_convergent_in x0 : ( ( ) (
V30()
real ext-real )
Real) &
lim_right (
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,) ,
x0 : ( ( ) (
V30()
real ext-real )
Real) ) : ( ( ) (
V30()
real ext-real )
Real)
= lim_right (
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,) ,
x0 : ( ( ) (
V30()
real ext-real )
Real) ) : ( ( ) (
V30()
real ext-real )
Real) & ( for
r being ( ( ) (
V30()
real ext-real )
Real) st
x0 : ( ( ) (
V30()
real ext-real )
Real)
< r : ( ( ) (
V30()
real ext-real )
Real) holds
ex
g being ( ( ) (
V30()
real ext-real )
Real) st
(
g : ( ( ) (
V30()
real ext-real )
Real)
< r : ( ( ) (
V30()
real ext-real )
Real) &
x0 : ( ( ) (
V30()
real ext-real )
Real)
< g : ( ( ) (
V30()
real ext-real )
Real) &
g : ( ( ) (
V30()
real ext-real )
Real)
in dom f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) ) ) & ex
r being ( ( ) (
V30()
real ext-real )
Real) st
(
0 : ( ( ) (
empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V30()
real ext-real non
positive non
negative V71()
V72()
V73()
V74()
V75()
V76()
V77()
V87()
V88() )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) )
< r : ( ( ) (
V30()
real ext-real )
Real) & ( for
g being ( ( ) (
V30()
real ext-real )
Real) st
g : ( ( ) (
V30()
real ext-real )
Real)
in (dom f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) )
/\ ].x0 : ( ( ) ( V30() real ext-real ) Real) ,(x0 : ( ( ) ( V30() real ext-real ) Real) + r : ( ( ) ( V30() real ext-real ) Real) ) : ( ( ) ( V30() real ext-real ) Element of REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) .[ : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) holds
(
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
. g : ( ( ) (
V30()
real ext-real )
Real) : ( ( ) (
V30()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
<= f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
. g : ( ( ) (
V30()
real ext-real )
Real) : ( ( ) (
V30()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) &
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
. g : ( ( ) (
V30()
real ext-real )
Real) : ( ( ) (
V30()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
<= f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
. g : ( ( ) (
V30()
real ext-real )
Real) : ( ( ) (
V30()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) ) ) & ( (
(dom f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) )
/\ ].x0 : ( ( ) ( V30() real ext-real ) Real) ,(x0 : ( ( ) ( V30() real ext-real ) Real) + r : ( ( ) ( V30() real ext-real ) Real) ) : ( ( ) ( V30() real ext-real ) Element of REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) .[ : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) )
c= (dom f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) )
/\ ].x0 : ( ( ) ( V30() real ext-real ) Real) ,(x0 : ( ( ) ( V30() real ext-real ) Real) + r : ( ( ) ( V30() real ext-real ) Real) ) : ( ( ) ( V30() real ext-real ) Element of REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) .[ : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) &
(dom f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) )
/\ ].x0 : ( ( ) ( V30() real ext-real ) Real) ,(x0 : ( ( ) ( V30() real ext-real ) Real) + r : ( ( ) ( V30() real ext-real ) Real) ) : ( ( ) ( V30() real ext-real ) Element of REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) .[ : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) )
c= (dom f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) )
/\ ].x0 : ( ( ) ( V30() real ext-real ) Real) ,(x0 : ( ( ) ( V30() real ext-real ) Real) + r : ( ( ) ( V30() real ext-real ) Real) ) : ( ( ) ( V30() real ext-real ) Element of REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) .[ : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) ) or (
(dom f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) )
/\ ].x0 : ( ( ) ( V30() real ext-real ) Real) ,(x0 : ( ( ) ( V30() real ext-real ) Real) + r : ( ( ) ( V30() real ext-real ) Real) ) : ( ( ) ( V30() real ext-real ) Element of REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) .[ : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) )
c= (dom f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) )
/\ ].x0 : ( ( ) ( V30() real ext-real ) Real) ,(x0 : ( ( ) ( V30() real ext-real ) Real) + r : ( ( ) ( V30() real ext-real ) Real) ) : ( ( ) ( V30() real ext-real ) Element of REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) .[ : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) &
(dom f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) )
/\ ].x0 : ( ( ) ( V30() real ext-real ) Real) ,(x0 : ( ( ) ( V30() real ext-real ) Real) + r : ( ( ) ( V30() real ext-real ) Real) ) : ( ( ) ( V30() real ext-real ) Element of REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) .[ : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) )
c= (dom f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) )
/\ ].x0 : ( ( ) ( V30() real ext-real ) Real) ,(x0 : ( ( ) ( V30() real ext-real ) Real) + r : ( ( ) ( V30() real ext-real ) Real) ) : ( ( ) ( V30() real ext-real ) Element of REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) .[ : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) ) ) ) holds
(
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
is_right_convergent_in x0 : ( ( ) (
V30()
real ext-real )
Real) &
lim_right (
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,) ,
x0 : ( ( ) (
V30()
real ext-real )
Real) ) : ( ( ) (
V30()
real ext-real )
Real)
= lim_right (
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,) ,
x0 : ( ( ) (
V30()
real ext-real )
Real) ) : ( ( ) (
V30()
real ext-real )
Real) ) ;
theorem
for
x0 being ( ( ) (
V30()
real ext-real )
Real)
for
f1,
f2,
f being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,) st
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
is_right_convergent_in x0 : ( ( ) (
V30()
real ext-real )
Real) &
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
is_right_convergent_in x0 : ( ( ) (
V30()
real ext-real )
Real) &
lim_right (
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,) ,
x0 : ( ( ) (
V30()
real ext-real )
Real) ) : ( ( ) (
V30()
real ext-real )
Real)
= lim_right (
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,) ,
x0 : ( ( ) (
V30()
real ext-real )
Real) ) : ( ( ) (
V30()
real ext-real )
Real) & ex
r being ( ( ) (
V30()
real ext-real )
Real) st
(
0 : ( ( ) (
empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V30()
real ext-real non
positive non
negative V71()
V72()
V73()
V74()
V75()
V76()
V77()
V87()
V88() )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) )
< r : ( ( ) (
V30()
real ext-real )
Real) &
].x0 : ( ( ) ( V30() real ext-real ) Real) ,(x0 : ( ( ) ( V30() real ext-real ) Real) + r : ( ( ) ( V30() real ext-real ) Real) ) : ( ( ) ( V30() real ext-real ) Element of REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) .[ : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) )
c= ((dom f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( ( ) ( V71() V72() V73() ) Element of K19(REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) : ( ( ) ( ) set ) ) /\ (dom f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( ( ) ( V71() V72() V73() ) Element of K19(REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) )
/\ (dom f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) & ( for
g being ( ( ) (
V30()
real ext-real )
Real) st
g : ( ( ) (
V30()
real ext-real )
Real)
in ].x0 : ( ( ) ( V30() real ext-real ) Real) ,(x0 : ( ( ) ( V30() real ext-real ) Real) + r : ( ( ) ( V30() real ext-real ) Real) ) : ( ( ) ( V30() real ext-real ) Element of REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) .[ : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) holds
(
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
. g : ( ( ) (
V30()
real ext-real )
Real) : ( ( ) (
V30()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
<= f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
. g : ( ( ) (
V30()
real ext-real )
Real) : ( ( ) (
V30()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) &
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
. g : ( ( ) (
V30()
real ext-real )
Real) : ( ( ) (
V30()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
<= f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
. g : ( ( ) (
V30()
real ext-real )
Real) : ( ( ) (
V30()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) ) ) ) holds
(
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
is_right_convergent_in x0 : ( ( ) (
V30()
real ext-real )
Real) &
lim_right (
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,) ,
x0 : ( ( ) (
V30()
real ext-real )
Real) ) : ( ( ) (
V30()
real ext-real )
Real)
= lim_right (
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,) ,
x0 : ( ( ) (
V30()
real ext-real )
Real) ) : ( ( ) (
V30()
real ext-real )
Real) ) ;
theorem
for
x0 being ( ( ) (
V30()
real ext-real )
Real)
for
f1,
f2 being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,) st
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
is_left_convergent_in x0 : ( ( ) (
V30()
real ext-real )
Real) &
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
is_left_convergent_in x0 : ( ( ) (
V30()
real ext-real )
Real) & ex
r being ( ( ) (
V30()
real ext-real )
Real) st
(
0 : ( ( ) (
empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V30()
real ext-real non
positive non
negative V71()
V72()
V73()
V74()
V75()
V76()
V77()
V87()
V88() )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) )
< r : ( ( ) (
V30()
real ext-real )
Real) & ( (
(dom f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) )
/\ ].(x0 : ( ( ) ( V30() real ext-real ) Real) - r : ( ( ) ( V30() real ext-real ) Real) ) : ( ( ) ( V30() real ext-real ) Element of REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) ,x0 : ( ( ) ( V30() real ext-real ) Real) .[ : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) )
c= (dom f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) )
/\ ].(x0 : ( ( ) ( V30() real ext-real ) Real) - r : ( ( ) ( V30() real ext-real ) Real) ) : ( ( ) ( V30() real ext-real ) Element of REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) ,x0 : ( ( ) ( V30() real ext-real ) Real) .[ : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) & ( for
g being ( ( ) (
V30()
real ext-real )
Real) st
g : ( ( ) (
V30()
real ext-real )
Real)
in (dom f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) )
/\ ].(x0 : ( ( ) ( V30() real ext-real ) Real) - r : ( ( ) ( V30() real ext-real ) Real) ) : ( ( ) ( V30() real ext-real ) Element of REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) ,x0 : ( ( ) ( V30() real ext-real ) Real) .[ : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) holds
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
. g : ( ( ) (
V30()
real ext-real )
Real) : ( ( ) (
V30()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
<= f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
. g : ( ( ) (
V30()
real ext-real )
Real) : ( ( ) (
V30()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) ) ) or (
(dom f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) )
/\ ].(x0 : ( ( ) ( V30() real ext-real ) Real) - r : ( ( ) ( V30() real ext-real ) Real) ) : ( ( ) ( V30() real ext-real ) Element of REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) ,x0 : ( ( ) ( V30() real ext-real ) Real) .[ : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) )
c= (dom f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) )
/\ ].(x0 : ( ( ) ( V30() real ext-real ) Real) - r : ( ( ) ( V30() real ext-real ) Real) ) : ( ( ) ( V30() real ext-real ) Element of REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) ,x0 : ( ( ) ( V30() real ext-real ) Real) .[ : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) & ( for
g being ( ( ) (
V30()
real ext-real )
Real) st
g : ( ( ) (
V30()
real ext-real )
Real)
in (dom f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) )
/\ ].(x0 : ( ( ) ( V30() real ext-real ) Real) - r : ( ( ) ( V30() real ext-real ) Real) ) : ( ( ) ( V30() real ext-real ) Element of REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) ,x0 : ( ( ) ( V30() real ext-real ) Real) .[ : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) holds
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
. g : ( ( ) (
V30()
real ext-real )
Real) : ( ( ) (
V30()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
<= f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
. g : ( ( ) (
V30()
real ext-real )
Real) : ( ( ) (
V30()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) ) ) ) ) holds
lim_left (
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,) ,
x0 : ( ( ) (
V30()
real ext-real )
Real) ) : ( ( ) (
V30()
real ext-real )
Real)
<= lim_left (
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,) ,
x0 : ( ( ) (
V30()
real ext-real )
Real) ) : ( ( ) (
V30()
real ext-real )
Real) ;
theorem
for
x0 being ( ( ) (
V30()
real ext-real )
Real)
for
f1,
f2 being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,) st
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
is_right_convergent_in x0 : ( ( ) (
V30()
real ext-real )
Real) &
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
is_right_convergent_in x0 : ( ( ) (
V30()
real ext-real )
Real) & ex
r being ( ( ) (
V30()
real ext-real )
Real) st
(
0 : ( ( ) (
empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V30()
real ext-real non
positive non
negative V71()
V72()
V73()
V74()
V75()
V76()
V77()
V87()
V88() )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) )
< r : ( ( ) (
V30()
real ext-real )
Real) & ( (
(dom f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) )
/\ ].x0 : ( ( ) ( V30() real ext-real ) Real) ,(x0 : ( ( ) ( V30() real ext-real ) Real) + r : ( ( ) ( V30() real ext-real ) Real) ) : ( ( ) ( V30() real ext-real ) Element of REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) .[ : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) )
c= (dom f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) )
/\ ].x0 : ( ( ) ( V30() real ext-real ) Real) ,(x0 : ( ( ) ( V30() real ext-real ) Real) + r : ( ( ) ( V30() real ext-real ) Real) ) : ( ( ) ( V30() real ext-real ) Element of REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) .[ : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) & ( for
g being ( ( ) (
V30()
real ext-real )
Real) st
g : ( ( ) (
V30()
real ext-real )
Real)
in (dom f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) )
/\ ].x0 : ( ( ) ( V30() real ext-real ) Real) ,(x0 : ( ( ) ( V30() real ext-real ) Real) + r : ( ( ) ( V30() real ext-real ) Real) ) : ( ( ) ( V30() real ext-real ) Element of REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) .[ : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) holds
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
. g : ( ( ) (
V30()
real ext-real )
Real) : ( ( ) (
V30()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
<= f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
. g : ( ( ) (
V30()
real ext-real )
Real) : ( ( ) (
V30()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) ) ) or (
(dom f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) )
/\ ].x0 : ( ( ) ( V30() real ext-real ) Real) ,(x0 : ( ( ) ( V30() real ext-real ) Real) + r : ( ( ) ( V30() real ext-real ) Real) ) : ( ( ) ( V30() real ext-real ) Element of REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) .[ : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) )
c= (dom f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) )
/\ ].x0 : ( ( ) ( V30() real ext-real ) Real) ,(x0 : ( ( ) ( V30() real ext-real ) Real) + r : ( ( ) ( V30() real ext-real ) Real) ) : ( ( ) ( V30() real ext-real ) Element of REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) .[ : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) & ( for
g being ( ( ) (
V30()
real ext-real )
Real) st
g : ( ( ) (
V30()
real ext-real )
Real)
in (dom f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) )
/\ ].x0 : ( ( ) ( V30() real ext-real ) Real) ,(x0 : ( ( ) ( V30() real ext-real ) Real) + r : ( ( ) ( V30() real ext-real ) Real) ) : ( ( ) ( V30() real ext-real ) Element of REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) .[ : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) holds
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
. g : ( ( ) (
V30()
real ext-real )
Real) : ( ( ) (
V30()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
<= f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
. g : ( ( ) (
V30()
real ext-real )
Real) : ( ( ) (
V30()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) ) ) ) ) holds
lim_right (
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,) ,
x0 : ( ( ) (
V30()
real ext-real )
Real) ) : ( ( ) (
V30()
real ext-real )
Real)
<= lim_right (
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,) ,
x0 : ( ( ) (
V30()
real ext-real )
Real) ) : ( ( ) (
V30()
real ext-real )
Real) ;
theorem
for
x0 being ( ( ) (
V30()
real ext-real )
Real)
for
f being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,) st (
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
is_left_divergent_to+infty_in x0 : ( ( ) (
V30()
real ext-real )
Real) or
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
is_left_divergent_to-infty_in x0 : ( ( ) (
V30()
real ext-real )
Real) ) & ( for
r being ( ( ) (
V30()
real ext-real )
Real) st
r : ( ( ) (
V30()
real ext-real )
Real)
< x0 : ( ( ) (
V30()
real ext-real )
Real) holds
ex
g being ( ( ) (
V30()
real ext-real )
Real) st
(
r : ( ( ) (
V30()
real ext-real )
Real)
< g : ( ( ) (
V30()
real ext-real )
Real) &
g : ( ( ) (
V30()
real ext-real )
Real)
< x0 : ( ( ) (
V30()
real ext-real )
Real) &
g : ( ( ) (
V30()
real ext-real )
Real)
in dom f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) &
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
. g : ( ( ) (
V30()
real ext-real )
Real) : ( ( ) (
V30()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
<> 0 : ( ( ) (
empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V30()
real ext-real non
positive non
negative V71()
V72()
V73()
V74()
V75()
V76()
V77()
V87()
V88() )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) ) ) ) holds
(
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
^ : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
is_left_convergent_in x0 : ( ( ) (
V30()
real ext-real )
Real) &
lim_left (
(f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) ^) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) ,
x0 : ( ( ) (
V30()
real ext-real )
Real) ) : ( ( ) (
V30()
real ext-real )
Real)
= 0 : ( ( ) (
empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V30()
real ext-real non
positive non
negative V71()
V72()
V73()
V74()
V75()
V76()
V77()
V87()
V88() )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) ) ) ;
theorem
for
x0 being ( ( ) (
V30()
real ext-real )
Real)
for
f being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,) st (
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
is_right_divergent_to+infty_in x0 : ( ( ) (
V30()
real ext-real )
Real) or
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
is_right_divergent_to-infty_in x0 : ( ( ) (
V30()
real ext-real )
Real) ) & ( for
r being ( ( ) (
V30()
real ext-real )
Real) st
x0 : ( ( ) (
V30()
real ext-real )
Real)
< r : ( ( ) (
V30()
real ext-real )
Real) holds
ex
g being ( ( ) (
V30()
real ext-real )
Real) st
(
g : ( ( ) (
V30()
real ext-real )
Real)
< r : ( ( ) (
V30()
real ext-real )
Real) &
x0 : ( ( ) (
V30()
real ext-real )
Real)
< g : ( ( ) (
V30()
real ext-real )
Real) &
g : ( ( ) (
V30()
real ext-real )
Real)
in dom f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) &
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
. g : ( ( ) (
V30()
real ext-real )
Real) : ( ( ) (
V30()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
<> 0 : ( ( ) (
empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V30()
real ext-real non
positive non
negative V71()
V72()
V73()
V74()
V75()
V76()
V77()
V87()
V88() )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) ) ) ) holds
(
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
^ : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
is_right_convergent_in x0 : ( ( ) (
V30()
real ext-real )
Real) &
lim_right (
(f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) ^) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) ) ,
x0 : ( ( ) (
V30()
real ext-real )
Real) ) : ( ( ) (
V30()
real ext-real )
Real)
= 0 : ( ( ) (
empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V30()
real ext-real non
positive non
negative V71()
V72()
V73()
V74()
V75()
V76()
V77()
V87()
V88() )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) ) ) ;
theorem
for
x0 being ( ( ) (
V30()
real ext-real )
Real)
for
f being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,) st
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
is_left_convergent_in x0 : ( ( ) (
V30()
real ext-real )
Real) &
lim_left (
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,) ,
x0 : ( ( ) (
V30()
real ext-real )
Real) ) : ( ( ) (
V30()
real ext-real )
Real)
= 0 : ( ( ) (
empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V30()
real ext-real non
positive non
negative V71()
V72()
V73()
V74()
V75()
V76()
V77()
V87()
V88() )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) ) & ( for
r being ( ( ) (
V30()
real ext-real )
Real) st
r : ( ( ) (
V30()
real ext-real )
Real)
< x0 : ( ( ) (
V30()
real ext-real )
Real) holds
ex
g being ( ( ) (
V30()
real ext-real )
Real) st
(
r : ( ( ) (
V30()
real ext-real )
Real)
< g : ( ( ) (
V30()
real ext-real )
Real) &
g : ( ( ) (
V30()
real ext-real )
Real)
< x0 : ( ( ) (
V30()
real ext-real )
Real) &
g : ( ( ) (
V30()
real ext-real )
Real)
in dom f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) &
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
. g : ( ( ) (
V30()
real ext-real )
Real) : ( ( ) (
V30()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
<> 0 : ( ( ) (
empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V30()
real ext-real non
positive non
negative V71()
V72()
V73()
V74()
V75()
V76()
V77()
V87()
V88() )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) ) ) ) & ex
r being ( ( ) (
V30()
real ext-real )
Real) st
(
0 : ( ( ) (
empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V30()
real ext-real non
positive non
negative V71()
V72()
V73()
V74()
V75()
V76()
V77()
V87()
V88() )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) )
< r : ( ( ) (
V30()
real ext-real )
Real) & ( for
g being ( ( ) (
V30()
real ext-real )
Real) st
g : ( ( ) (
V30()
real ext-real )
Real)
in (dom f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) )
/\ ].(x0 : ( ( ) ( V30() real ext-real ) Real) - r : ( ( ) ( V30() real ext-real ) Real) ) : ( ( ) ( V30() real ext-real ) Element of REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) ,x0 : ( ( ) ( V30() real ext-real ) Real) .[ : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) holds
0 : ( ( ) (
empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V30()
real ext-real non
positive non
negative V71()
V72()
V73()
V74()
V75()
V76()
V77()
V87()
V88() )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) )
<= f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
. g : ( ( ) (
V30()
real ext-real )
Real) : ( ( ) (
V30()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) ) ) holds
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
^ : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
is_left_divergent_to+infty_in x0 : ( ( ) (
V30()
real ext-real )
Real) ;
theorem
for
x0 being ( ( ) (
V30()
real ext-real )
Real)
for
f being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,) st
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
is_left_convergent_in x0 : ( ( ) (
V30()
real ext-real )
Real) &
lim_left (
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,) ,
x0 : ( ( ) (
V30()
real ext-real )
Real) ) : ( ( ) (
V30()
real ext-real )
Real)
= 0 : ( ( ) (
empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V30()
real ext-real non
positive non
negative V71()
V72()
V73()
V74()
V75()
V76()
V77()
V87()
V88() )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) ) & ( for
r being ( ( ) (
V30()
real ext-real )
Real) st
r : ( ( ) (
V30()
real ext-real )
Real)
< x0 : ( ( ) (
V30()
real ext-real )
Real) holds
ex
g being ( ( ) (
V30()
real ext-real )
Real) st
(
r : ( ( ) (
V30()
real ext-real )
Real)
< g : ( ( ) (
V30()
real ext-real )
Real) &
g : ( ( ) (
V30()
real ext-real )
Real)
< x0 : ( ( ) (
V30()
real ext-real )
Real) &
g : ( ( ) (
V30()
real ext-real )
Real)
in dom f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) &
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
. g : ( ( ) (
V30()
real ext-real )
Real) : ( ( ) (
V30()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
<> 0 : ( ( ) (
empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V30()
real ext-real non
positive non
negative V71()
V72()
V73()
V74()
V75()
V76()
V77()
V87()
V88() )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) ) ) ) & ex
r being ( ( ) (
V30()
real ext-real )
Real) st
(
0 : ( ( ) (
empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V30()
real ext-real non
positive non
negative V71()
V72()
V73()
V74()
V75()
V76()
V77()
V87()
V88() )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) )
< r : ( ( ) (
V30()
real ext-real )
Real) & ( for
g being ( ( ) (
V30()
real ext-real )
Real) st
g : ( ( ) (
V30()
real ext-real )
Real)
in (dom f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) )
/\ ].(x0 : ( ( ) ( V30() real ext-real ) Real) - r : ( ( ) ( V30() real ext-real ) Real) ) : ( ( ) ( V30() real ext-real ) Element of REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) ,x0 : ( ( ) ( V30() real ext-real ) Real) .[ : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) holds
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
. g : ( ( ) (
V30()
real ext-real )
Real) : ( ( ) (
V30()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
<= 0 : ( ( ) (
empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V30()
real ext-real non
positive non
negative V71()
V72()
V73()
V74()
V75()
V76()
V77()
V87()
V88() )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) ) ) ) holds
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
^ : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
is_left_divergent_to-infty_in x0 : ( ( ) (
V30()
real ext-real )
Real) ;
theorem
for
x0 being ( ( ) (
V30()
real ext-real )
Real)
for
f being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,) st
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
is_right_convergent_in x0 : ( ( ) (
V30()
real ext-real )
Real) &
lim_right (
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,) ,
x0 : ( ( ) (
V30()
real ext-real )
Real) ) : ( ( ) (
V30()
real ext-real )
Real)
= 0 : ( ( ) (
empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V30()
real ext-real non
positive non
negative V71()
V72()
V73()
V74()
V75()
V76()
V77()
V87()
V88() )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) ) & ( for
r being ( ( ) (
V30()
real ext-real )
Real) st
x0 : ( ( ) (
V30()
real ext-real )
Real)
< r : ( ( ) (
V30()
real ext-real )
Real) holds
ex
g being ( ( ) (
V30()
real ext-real )
Real) st
(
g : ( ( ) (
V30()
real ext-real )
Real)
< r : ( ( ) (
V30()
real ext-real )
Real) &
x0 : ( ( ) (
V30()
real ext-real )
Real)
< g : ( ( ) (
V30()
real ext-real )
Real) &
g : ( ( ) (
V30()
real ext-real )
Real)
in dom f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) &
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
. g : ( ( ) (
V30()
real ext-real )
Real) : ( ( ) (
V30()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
<> 0 : ( ( ) (
empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V30()
real ext-real non
positive non
negative V71()
V72()
V73()
V74()
V75()
V76()
V77()
V87()
V88() )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) ) ) ) & ex
r being ( ( ) (
V30()
real ext-real )
Real) st
(
0 : ( ( ) (
empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V30()
real ext-real non
positive non
negative V71()
V72()
V73()
V74()
V75()
V76()
V77()
V87()
V88() )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) )
< r : ( ( ) (
V30()
real ext-real )
Real) & ( for
g being ( ( ) (
V30()
real ext-real )
Real) st
g : ( ( ) (
V30()
real ext-real )
Real)
in (dom f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) )
/\ ].x0 : ( ( ) ( V30() real ext-real ) Real) ,(x0 : ( ( ) ( V30() real ext-real ) Real) + r : ( ( ) ( V30() real ext-real ) Real) ) : ( ( ) ( V30() real ext-real ) Element of REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) .[ : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) holds
0 : ( ( ) (
empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V30()
real ext-real non
positive non
negative V71()
V72()
V73()
V74()
V75()
V76()
V77()
V87()
V88() )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) )
<= f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
. g : ( ( ) (
V30()
real ext-real )
Real) : ( ( ) (
V30()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) ) ) holds
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
^ : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
is_right_divergent_to+infty_in x0 : ( ( ) (
V30()
real ext-real )
Real) ;
theorem
for
x0 being ( ( ) (
V30()
real ext-real )
Real)
for
f being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,) st
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
is_right_convergent_in x0 : ( ( ) (
V30()
real ext-real )
Real) &
lim_right (
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,) ,
x0 : ( ( ) (
V30()
real ext-real )
Real) ) : ( ( ) (
V30()
real ext-real )
Real)
= 0 : ( ( ) (
empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V30()
real ext-real non
positive non
negative V71()
V72()
V73()
V74()
V75()
V76()
V77()
V87()
V88() )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) ) & ( for
r being ( ( ) (
V30()
real ext-real )
Real) st
x0 : ( ( ) (
V30()
real ext-real )
Real)
< r : ( ( ) (
V30()
real ext-real )
Real) holds
ex
g being ( ( ) (
V30()
real ext-real )
Real) st
(
g : ( ( ) (
V30()
real ext-real )
Real)
< r : ( ( ) (
V30()
real ext-real )
Real) &
x0 : ( ( ) (
V30()
real ext-real )
Real)
< g : ( ( ) (
V30()
real ext-real )
Real) &
g : ( ( ) (
V30()
real ext-real )
Real)
in dom f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) &
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
. g : ( ( ) (
V30()
real ext-real )
Real) : ( ( ) (
V30()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
<> 0 : ( ( ) (
empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V30()
real ext-real non
positive non
negative V71()
V72()
V73()
V74()
V75()
V76()
V77()
V87()
V88() )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) ) ) ) & ex
r being ( ( ) (
V30()
real ext-real )
Real) st
(
0 : ( ( ) (
empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V30()
real ext-real non
positive non
negative V71()
V72()
V73()
V74()
V75()
V76()
V77()
V87()
V88() )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) )
< r : ( ( ) (
V30()
real ext-real )
Real) & ( for
g being ( ( ) (
V30()
real ext-real )
Real) st
g : ( ( ) (
V30()
real ext-real )
Real)
in (dom f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) )
/\ ].x0 : ( ( ) ( V30() real ext-real ) Real) ,(x0 : ( ( ) ( V30() real ext-real ) Real) + r : ( ( ) ( V30() real ext-real ) Real) ) : ( ( ) ( V30() real ext-real ) Element of REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() ) set ) ) .[ : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) holds
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
. g : ( ( ) (
V30()
real ext-real )
Real) : ( ( ) (
V30()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
<= 0 : ( ( ) (
empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V30()
real ext-real non
positive non
negative V71()
V72()
V73()
V74()
V75()
V76()
V77()
V87()
V88() )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) ( )
set ) ) ) ) ) holds
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
PartFunc of ,)
^ : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) )
V6()
complex-valued ext-real-valued real-valued )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77() )
set ) ) : ( ( ) (
complex-valued ext-real-valued real-valued )
set ) ) : ( ( ) ( )
set ) )
is_right_divergent_to-infty_in x0 : ( ( ) (
V30()
real ext-real )
Real) ;