begin
theorem
for
f being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V6()
V35()
V36()
V37() )
PartFunc of ,)
for
x0,
t being ( ( ) (
V30()
real ext-real )
Real) holds
(
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V6()
V35()
V36()
V37() )
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()
V101()
V102()
V104() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V6()
V35()
V36()
V37() )
PartFunc of ,) ,
x0 : ( ( ) (
V30()
real ext-real )
Real) ) : ( ( ) (
V30()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
= t : ( ( ) (
V30()
real ext-real )
Real) iff ( ( for
r being ( ( ) (
V30()
real ext-real )
Real) st
x0 : ( ( ) (
V30()
real ext-real )
Real)
< r : ( (
V6()
quasi_total ) (
V1()
V4(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V99()
V101() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ) : ( ( ) ( )
set ) ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V6() non
empty total quasi_total V35()
V36()
V37() )
Real_Sequence) holds
ex
t being ( ( ) (
V30()
real ext-real )
Real) st
(
t : ( ( ) (
V30()
real ext-real )
Real)
< r : ( (
V6()
quasi_total ) (
V1()
V4(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V99()
V101() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ) : ( ( ) ( )
set ) ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V6() non
empty total quasi_total V35()
V36()
V37() )
Real_Sequence) &
x0 : ( ( ) (
V30()
real ext-real )
Real)
< t : ( ( ) (
V30()
real ext-real )
Real) &
t : ( ( ) (
V30()
real ext-real )
Real)
in dom f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V6()
V35()
V36()
V37() )
PartFunc of ,) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ) : ( ( ) ( )
set ) ) ) ) & ( for
a being ( (
V6()
quasi_total ) (
V1()
V4(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V99()
V101() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ) : ( ( ) ( )
set ) ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V6() non
empty total quasi_total V35()
V36()
V37() )
Real_Sequence) st
a : ( (
V6()
quasi_total ) (
V1()
V4(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V99()
V101() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ) : ( ( ) ( )
set ) ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V6() non
empty total quasi_total V35()
V36()
V37() )
Real_Sequence) is
convergent &
lim a : ( (
V6()
quasi_total ) (
V1()
V4(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V99()
V101() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ) : ( ( ) ( )
set ) ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V6() non
empty total quasi_total V35()
V36()
V37() )
Real_Sequence) : ( ( ) (
V30()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
= x0 : ( ( ) (
V30()
real ext-real )
Real) &
rng a : ( (
V6()
quasi_total ) (
V1()
V4(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V99()
V101() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ) : ( ( ) ( )
set ) ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V6() non
empty total quasi_total V35()
V36()
V37() )
Real_Sequence) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ) : ( ( ) ( )
set ) )
c= (dom f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) V6() V35() V36() V37() ) PartFunc of ,) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ) : ( ( ) ( )
set ) )
/\ (right_open_halfline x0 : ( ( ) ( V30() real ext-real ) Real) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ) : ( ( ) ( )
set ) ) holds
(
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V6()
V35()
V36()
V37() )
PartFunc of ,)
/* a : ( (
V6()
quasi_total ) (
V1()
V4(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V99()
V101() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ) : ( ( ) ( )
set ) ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V6() non
empty total quasi_total V35()
V36()
V37() )
Real_Sequence) : ( (
V6()
quasi_total ) (
V1()
V4(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V99()
V101() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ) : ( ( ) ( )
set ) ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V6() non
empty total quasi_total V35()
V36()
V37() )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V99()
V101() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ) : ( ( ) (
V35()
V36()
V37() )
set ) ) : ( ( ) ( )
set ) ) is
convergent &
lim (f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) V6() V35() V36() V37() ) PartFunc of ,) /* a : ( ( V6() quasi_total ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V99() V101() ) Element of K19(REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) : ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) V6() non empty total quasi_total V35() V36() V37() ) Real_Sequence) ) : ( (
V6()
quasi_total ) (
V1()
V4(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V99()
V101() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ) : ( ( ) ( )
set ) ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V6() non
empty total quasi_total V35()
V36()
V37() )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V99()
V101() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ) : ( ( ) (
V35()
V36()
V37() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V30()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
= t : ( ( ) (
V30()
real ext-real )
Real) ) ) ) ) ;
theorem
for
f being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V6()
V35()
V36()
V37() )
PartFunc of ,)
for
x0,
t being ( ( ) (
V30()
real ext-real )
Real) holds
(
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V6()
V35()
V36()
V37() )
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()
V101()
V102()
V104() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V6()
V35()
V36()
V37() )
PartFunc of ,) ,
x0 : ( ( ) (
V30()
real ext-real )
Real) ) : ( ( ) (
V30()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
= t : ( ( ) (
V30()
real ext-real )
Real) iff ( ( for
r being ( ( ) (
V30()
real ext-real )
Real) st
r : ( (
V6()
quasi_total ) (
V1()
V4(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V99()
V101() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ) : ( ( ) ( )
set ) ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V6() non
empty total quasi_total V35()
V36()
V37() )
Real_Sequence)
< x0 : ( ( ) (
V30()
real ext-real )
Real) holds
ex
t being ( ( ) (
V30()
real ext-real )
Real) st
(
r : ( (
V6()
quasi_total ) (
V1()
V4(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V99()
V101() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ) : ( ( ) ( )
set ) ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V6() non
empty total quasi_total V35()
V36()
V37() )
Real_Sequence)
< t : ( ( ) (
V30()
real ext-real )
Real) &
t : ( ( ) (
V30()
real ext-real )
Real)
< x0 : ( ( ) (
V30()
real ext-real )
Real) &
t : ( ( ) (
V30()
real ext-real )
Real)
in dom f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V6()
V35()
V36()
V37() )
PartFunc of ,) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ) : ( ( ) ( )
set ) ) ) ) & ( for
a being ( (
V6()
quasi_total ) (
V1()
V4(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V99()
V101() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ) : ( ( ) ( )
set ) ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V6() non
empty total quasi_total V35()
V36()
V37() )
Real_Sequence) st
a : ( (
V6()
quasi_total ) (
V1()
V4(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V99()
V101() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ) : ( ( ) ( )
set ) ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V6() non
empty total quasi_total V35()
V36()
V37() )
Real_Sequence) is
convergent &
lim a : ( (
V6()
quasi_total ) (
V1()
V4(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V99()
V101() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ) : ( ( ) ( )
set ) ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V6() non
empty total quasi_total V35()
V36()
V37() )
Real_Sequence) : ( ( ) (
V30()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
= x0 : ( ( ) (
V30()
real ext-real )
Real) &
rng a : ( (
V6()
quasi_total ) (
V1()
V4(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V99()
V101() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ) : ( ( ) ( )
set ) ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V6() non
empty total quasi_total V35()
V36()
V37() )
Real_Sequence) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ) : ( ( ) ( )
set ) )
c= (dom f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) V6() V35() V36() V37() ) PartFunc of ,) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ) : ( ( ) ( )
set ) )
/\ (left_open_halfline x0 : ( ( ) ( V30() real ext-real ) Real) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ) : ( ( ) ( )
set ) ) holds
(
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V6()
V35()
V36()
V37() )
PartFunc of ,)
/* a : ( (
V6()
quasi_total ) (
V1()
V4(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V99()
V101() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ) : ( ( ) ( )
set ) ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V6() non
empty total quasi_total V35()
V36()
V37() )
Real_Sequence) : ( (
V6()
quasi_total ) (
V1()
V4(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V99()
V101() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ) : ( ( ) ( )
set ) ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V6() non
empty total quasi_total V35()
V36()
V37() )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V99()
V101() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ) : ( ( ) (
V35()
V36()
V37() )
set ) ) : ( ( ) ( )
set ) ) is
convergent &
lim (f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) V6() V35() V36() V37() ) PartFunc of ,) /* a : ( ( V6() quasi_total ) ( V1() V4( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V71() V72() V73() V74() V75() V76() V77() V99() V101() ) Element of K19(REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) : ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) V6() non empty total quasi_total V35() V36() V37() ) Real_Sequence) ) : ( (
V6()
quasi_total ) (
V1()
V4(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V99()
V101() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ) : ( ( ) ( )
set ) ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V6() non
empty total quasi_total V35()
V36()
V37() )
Element of
K19(
K20(
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V99()
V101() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ) : ( ( ) (
V35()
V36()
V37() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V30()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
= t : ( ( ) (
V30()
real ext-real )
Real) ) ) ) ) ;
theorem
for
f,
g being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V6()
V35()
V36()
V37() )
PartFunc of ,)
for
x0 being ( ( ) (
V30()
real ext-real )
Real) st ex
N being ( ( ) (
open V71()
V72()
V73() )
Neighbourhood of
x0 : ( ( ) (
V30()
real ext-real )
Real) ) st
(
N : ( ( ) (
open V71()
V72()
V73() )
Neighbourhood of
b3 : ( ( ) (
V30()
real ext-real )
Real) )
c= dom f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V6()
V35()
V36()
V37() )
PartFunc of ,) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ) : ( ( ) ( )
set ) ) &
N : ( ( ) (
open V71()
V72()
V73() )
Neighbourhood of
b3 : ( ( ) (
V30()
real ext-real )
Real) )
c= dom g : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V6()
V35()
V36()
V37() )
PartFunc of ,) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ) : ( ( ) ( )
set ) ) &
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V6()
V35()
V36()
V37() )
PartFunc of ,)
is_differentiable_on N : ( ( ) (
open V71()
V72()
V73() )
Neighbourhood of
b3 : ( ( ) (
V30()
real ext-real )
Real) ) &
g : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V6()
V35()
V36()
V37() )
PartFunc of ,)
is_differentiable_on N : ( ( ) (
open V71()
V72()
V73() )
Neighbourhood of
b3 : ( ( ) (
V30()
real ext-real )
Real) ) &
N : ( ( ) (
open V71()
V72()
V73() )
Neighbourhood of
b3 : ( ( ) (
V30()
real ext-real )
Real) )
\ {x0 : ( ( ) ( V30() real ext-real ) Real) } : ( ( ) ( non
empty V71()
V72()
V73() )
set ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ) : ( ( ) ( )
set ) )
c= dom (f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) V6() V35() V36() V37() ) PartFunc of ,) / g : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) V6() V35() V36() V37() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V6()
V35()
V36()
V37() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ) : ( ( ) (
V35()
V36()
V37() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ) : ( ( ) ( )
set ) ) &
N : ( ( ) (
open V71()
V72()
V73() )
Neighbourhood of
b3 : ( ( ) (
V30()
real ext-real )
Real) )
c= dom ((f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) V6() V35() V36() V37() ) PartFunc of ,) `| N : ( ( ) ( open V71() V72() V73() ) Neighbourhood of b3 : ( ( ) ( V30() real ext-real ) Real) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) V6() V35() V36() V37() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ,REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) : ( ( ) ( V35() V36() V37() ) set ) ) : ( ( ) ( ) set ) ) / (g : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) V6() V35() V36() V37() ) PartFunc of ,) `| N : ( ( ) ( open V71() V72() V73() ) Neighbourhood of b3 : ( ( ) ( V30() real ext-real ) Real) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) V6() V35() V36() V37() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ,REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) : ( ( ) ( V35() V36() V37() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V6()
V35()
V36()
V37() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ) : ( ( ) (
V35()
V36()
V37() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ) : ( ( ) ( )
set ) ) &
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V6()
V35()
V36()
V37() )
PartFunc of ,)
. x0 : ( ( ) (
V30()
real ext-real )
Real) : ( ( ) (
V30()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
= 0 : ( ( ) (
empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V30()
real ext-real V71()
V72()
V73()
V74()
V75()
V76()
V77()
V93()
V94()
V101()
V104() )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V99()
V101() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ) : ( ( ) ( )
set ) ) ) &
g : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V6()
V35()
V36()
V37() )
PartFunc of ,)
. x0 : ( ( ) (
V30()
real ext-real )
Real) : ( ( ) (
V30()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
= 0 : ( ( ) (
empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V30()
real ext-real V71()
V72()
V73()
V74()
V75()
V76()
V77()
V93()
V94()
V101()
V104() )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V99()
V101() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ) : ( ( ) ( )
set ) ) ) &
(f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) V6() V35() V36() V37() ) PartFunc of ,) `| N : ( ( ) ( open V71() V72() V73() ) Neighbourhood of b3 : ( ( ) ( V30() real ext-real ) Real) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V6()
V35()
V36()
V37() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ) : ( ( ) (
V35()
V36()
V37() )
set ) ) : ( ( ) ( )
set ) )
/ (g : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) V6() V35() V36() V37() ) PartFunc of ,) `| N : ( ( ) ( open V71() V72() V73() ) Neighbourhood of b3 : ( ( ) ( V30() real ext-real ) Real) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V6()
V35()
V36()
V37() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ) : ( ( ) (
V35()
V36()
V37() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V6()
V35()
V36()
V37() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ) : ( ( ) (
V35()
V36()
V37() )
set ) ) : ( ( ) ( )
set ) )
is_divergent_to+infty_in x0 : ( ( ) (
V30()
real ext-real )
Real) ) holds
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V6()
V35()
V36()
V37() )
PartFunc of ,)
/ g : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V6()
V35()
V36()
V37() )
PartFunc of ,) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V6()
V35()
V36()
V37() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ) : ( ( ) (
V35()
V36()
V37() )
set ) ) : ( ( ) ( )
set ) )
is_divergent_to+infty_in x0 : ( ( ) (
V30()
real ext-real )
Real) ;
theorem
for
f,
g being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V6()
V35()
V36()
V37() )
PartFunc of ,)
for
x0 being ( ( ) (
V30()
real ext-real )
Real) st ex
N being ( ( ) (
open V71()
V72()
V73() )
Neighbourhood of
x0 : ( ( ) (
V30()
real ext-real )
Real) ) st
(
N : ( ( ) (
open V71()
V72()
V73() )
Neighbourhood of
b3 : ( ( ) (
V30()
real ext-real )
Real) )
c= dom f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V6()
V35()
V36()
V37() )
PartFunc of ,) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ) : ( ( ) ( )
set ) ) &
N : ( ( ) (
open V71()
V72()
V73() )
Neighbourhood of
b3 : ( ( ) (
V30()
real ext-real )
Real) )
c= dom g : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V6()
V35()
V36()
V37() )
PartFunc of ,) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ) : ( ( ) ( )
set ) ) &
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V6()
V35()
V36()
V37() )
PartFunc of ,)
is_differentiable_on N : ( ( ) (
open V71()
V72()
V73() )
Neighbourhood of
b3 : ( ( ) (
V30()
real ext-real )
Real) ) &
g : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V6()
V35()
V36()
V37() )
PartFunc of ,)
is_differentiable_on N : ( ( ) (
open V71()
V72()
V73() )
Neighbourhood of
b3 : ( ( ) (
V30()
real ext-real )
Real) ) &
N : ( ( ) (
open V71()
V72()
V73() )
Neighbourhood of
b3 : ( ( ) (
V30()
real ext-real )
Real) )
\ {x0 : ( ( ) ( V30() real ext-real ) Real) } : ( ( ) ( non
empty V71()
V72()
V73() )
set ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ) : ( ( ) ( )
set ) )
c= dom (f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) V6() V35() V36() V37() ) PartFunc of ,) / g : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) V6() V35() V36() V37() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V6()
V35()
V36()
V37() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ) : ( ( ) (
V35()
V36()
V37() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ) : ( ( ) ( )
set ) ) &
N : ( ( ) (
open V71()
V72()
V73() )
Neighbourhood of
b3 : ( ( ) (
V30()
real ext-real )
Real) )
c= dom ((f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) V6() V35() V36() V37() ) PartFunc of ,) `| N : ( ( ) ( open V71() V72() V73() ) Neighbourhood of b3 : ( ( ) ( V30() real ext-real ) Real) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) V6() V35() V36() V37() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ,REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) : ( ( ) ( V35() V36() V37() ) set ) ) : ( ( ) ( ) set ) ) / (g : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) V6() V35() V36() V37() ) PartFunc of ,) `| N : ( ( ) ( open V71() V72() V73() ) Neighbourhood of b3 : ( ( ) ( V30() real ext-real ) Real) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) V6() V35() V36() V37() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ,REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) : ( ( ) ( V35() V36() V37() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V6()
V35()
V36()
V37() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ) : ( ( ) (
V35()
V36()
V37() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ) : ( ( ) ( )
set ) ) &
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V6()
V35()
V36()
V37() )
PartFunc of ,)
. x0 : ( ( ) (
V30()
real ext-real )
Real) : ( ( ) (
V30()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
= 0 : ( ( ) (
empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V30()
real ext-real V71()
V72()
V73()
V74()
V75()
V76()
V77()
V93()
V94()
V101()
V104() )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V99()
V101() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ) : ( ( ) ( )
set ) ) ) &
g : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V6()
V35()
V36()
V37() )
PartFunc of ,)
. x0 : ( ( ) (
V30()
real ext-real )
Real) : ( ( ) (
V30()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
= 0 : ( ( ) (
empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V30()
real ext-real V71()
V72()
V73()
V74()
V75()
V76()
V77()
V93()
V94()
V101()
V104() )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V99()
V101() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ) : ( ( ) ( )
set ) ) ) &
(f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) V6() V35() V36() V37() ) PartFunc of ,) `| N : ( ( ) ( open V71() V72() V73() ) Neighbourhood of b3 : ( ( ) ( V30() real ext-real ) Real) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V6()
V35()
V36()
V37() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ) : ( ( ) (
V35()
V36()
V37() )
set ) ) : ( ( ) ( )
set ) )
/ (g : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) V6() V35() V36() V37() ) PartFunc of ,) `| N : ( ( ) ( open V71() V72() V73() ) Neighbourhood of b3 : ( ( ) ( V30() real ext-real ) Real) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V6()
V35()
V36()
V37() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ) : ( ( ) (
V35()
V36()
V37() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V6()
V35()
V36()
V37() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ) : ( ( ) (
V35()
V36()
V37() )
set ) ) : ( ( ) ( )
set ) )
is_divergent_to-infty_in x0 : ( ( ) (
V30()
real ext-real )
Real) ) holds
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V6()
V35()
V36()
V37() )
PartFunc of ,)
/ g : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V6()
V35()
V36()
V37() )
PartFunc of ,) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V6()
V35()
V36()
V37() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ) : ( ( ) (
V35()
V36()
V37() )
set ) ) : ( ( ) ( )
set ) )
is_divergent_to-infty_in x0 : ( ( ) (
V30()
real ext-real )
Real) ;
theorem
for
f,
g being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V6()
V35()
V36()
V37() )
PartFunc of ,)
for
x0 being ( ( ) (
V30()
real ext-real )
Real) st
x0 : ( ( ) (
V30()
real ext-real )
Real)
in (dom f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) V6() V35() V36() V37() ) PartFunc of ,) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ) : ( ( ) ( )
set ) )
/\ (dom g : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) V6() V35() V36() V37() ) PartFunc of ,) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ) : ( ( ) ( )
set ) ) & ex
r being ( ( ) (
V30()
real ext-real )
Real) st
(
r : ( ( ) (
V30()
real ext-real )
Real)
> 0 : ( ( ) (
empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V30()
real ext-real V71()
V72()
V73()
V74()
V75()
V76()
V77()
V93()
V94()
V101()
V104() )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V99()
V101() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
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() V101() V102() V104() ) set ) ) .] : ( ( ) (
closed V71()
V72()
V73()
V104() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ) : ( ( ) ( )
set ) )
c= dom f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V6()
V35()
V36()
V37() )
PartFunc of ,) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
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() V101() V102() V104() ) set ) ) .] : ( ( ) (
closed V71()
V72()
V73()
V104() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ) : ( ( ) ( )
set ) )
c= dom g : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V6()
V35()
V36()
V37() )
PartFunc of ,) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ) : ( ( ) ( )
set ) ) &
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V6()
V35()
V36()
V37() )
PartFunc of ,)
is_differentiable_on ].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() V101() V102() V104() ) set ) ) .[ : ( ( ) (
open V71()
V72()
V73()
V99()
V100()
V104() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ) : ( ( ) ( )
set ) ) &
g : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V6()
V35()
V36()
V37() )
PartFunc of ,)
is_differentiable_on ].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() V101() V102() V104() ) set ) ) .[ : ( ( ) (
open V71()
V72()
V73()
V99()
V100()
V104() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
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() V101() V102() V104() ) set ) ) .[ : ( ( ) (
open V71()
V72()
V73()
V99()
V100()
V104() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ) : ( ( ) ( )
set ) )
c= dom (f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) V6() V35() V36() V37() ) PartFunc of ,) / g : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) V6() V35() V36() V37() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V6()
V35()
V36()
V37() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ) : ( ( ) (
V35()
V36()
V37() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
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() V101() V102() V104() ) set ) ) .] : ( ( ) (
closed V71()
V72()
V73()
V104() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ) : ( ( ) ( )
set ) )
c= dom ((f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) V6() V35() V36() V37() ) 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() V101() V102() V104() ) set ) ) .[ : ( ( ) ( open V71() V72() V73() V99() V100() V104() ) Element of K19(REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) V6() V35() V36() V37() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ,REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) : ( ( ) ( V35() V36() V37() ) set ) ) : ( ( ) ( ) set ) ) / (g : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) V6() V35() V36() V37() ) 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() V101() V102() V104() ) set ) ) .[ : ( ( ) ( open V71() V72() V73() V99() V100() V104() ) Element of K19(REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) V6() V35() V36() V37() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ,REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) : ( ( ) ( V35() V36() V37() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V6()
V35()
V36()
V37() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ) : ( ( ) (
V35()
V36()
V37() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ) : ( ( ) ( )
set ) ) &
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V6()
V35()
V36()
V37() )
PartFunc of ,)
. x0 : ( ( ) (
V30()
real ext-real )
Real) : ( ( ) (
V30()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
= 0 : ( ( ) (
empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V30()
real ext-real V71()
V72()
V73()
V74()
V75()
V76()
V77()
V93()
V94()
V101()
V104() )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V99()
V101() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ) : ( ( ) ( )
set ) ) ) &
g : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V6()
V35()
V36()
V37() )
PartFunc of ,)
. x0 : ( ( ) (
V30()
real ext-real )
Real) : ( ( ) (
V30()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
= 0 : ( ( ) (
empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V30()
real ext-real V71()
V72()
V73()
V74()
V75()
V76()
V77()
V93()
V94()
V101()
V104() )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V99()
V101() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ) : ( ( ) ( )
set ) ) ) &
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V6()
V35()
V36()
V37() )
PartFunc of ,)
is_continuous_in x0 : ( ( ) (
V30()
real ext-real )
Real) &
g : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V6()
V35()
V36()
V37() )
PartFunc of ,)
is_continuous_in x0 : ( ( ) (
V30()
real ext-real )
Real) &
(f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) V6() V35() V36() V37() ) 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() V101() V102() V104() ) set ) ) .[ : ( ( ) ( open V71() V72() V73() V99() V100() V104() ) Element of K19(REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V6()
V35()
V36()
V37() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ) : ( ( ) (
V35()
V36()
V37() )
set ) ) : ( ( ) ( )
set ) )
/ (g : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) V6() V35() V36() V37() ) 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() V101() V102() V104() ) set ) ) .[ : ( ( ) ( open V71() V72() V73() V99() V100() V104() ) Element of K19(REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V6()
V35()
V36()
V37() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ) : ( ( ) (
V35()
V36()
V37() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V6()
V35()
V36()
V37() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ) : ( ( ) (
V35()
V36()
V37() )
set ) ) : ( ( ) ( )
set ) )
is_right_convergent_in x0 : ( ( ) (
V30()
real ext-real )
Real) ) holds
(
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V6()
V35()
V36()
V37() )
PartFunc of ,)
/ g : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V6()
V35()
V36()
V37() )
PartFunc of ,) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V6()
V35()
V36()
V37() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ) : ( ( ) (
V35()
V36()
V37() )
set ) ) : ( ( ) ( )
set ) )
is_right_convergent_in x0 : ( ( ) (
V30()
real ext-real )
Real) & ex
r being ( ( ) (
V30()
real ext-real )
Real) st
(
r : ( ( ) (
V30()
real ext-real )
Real)
> 0 : ( ( ) (
empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V30()
real ext-real V71()
V72()
V73()
V74()
V75()
V76()
V77()
V93()
V94()
V101()
V104() )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V99()
V101() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ) : ( ( ) ( )
set ) ) ) &
lim_right (
(f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) V6() V35() V36() V37() ) PartFunc of ,) / g : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) V6() V35() V36() V37() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V6()
V35()
V36()
V37() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ) : ( ( ) (
V35()
V36()
V37() )
set ) ) : ( ( ) ( )
set ) ) ,
x0 : ( ( ) (
V30()
real ext-real )
Real) ) : ( ( ) (
V30()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
= lim_right (
((f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) V6() V35() V36() V37() ) 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() V101() V102() V104() ) set ) ) .[ : ( ( ) ( open V71() V72() V73() V99() V100() V104() ) Element of K19(REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) V6() V35() V36() V37() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ,REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) : ( ( ) ( V35() V36() V37() ) set ) ) : ( ( ) ( ) set ) ) / (g : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) V6() V35() V36() V37() ) 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() V101() V102() V104() ) set ) ) .[ : ( ( ) ( open V71() V72() V73() V99() V100() V104() ) Element of K19(REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) V6() V35() V36() V37() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ,REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) : ( ( ) ( V35() V36() V37() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V6()
V35()
V36()
V37() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ) : ( ( ) (
V35()
V36()
V37() )
set ) ) : ( ( ) ( )
set ) ) ,
x0 : ( ( ) (
V30()
real ext-real )
Real) ) : ( ( ) (
V30()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ) ) ) ;
theorem
for
f,
g being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V6()
V35()
V36()
V37() )
PartFunc of ,)
for
x0 being ( ( ) (
V30()
real ext-real )
Real) st
x0 : ( ( ) (
V30()
real ext-real )
Real)
in (dom f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) V6() V35() V36() V37() ) PartFunc of ,) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ) : ( ( ) ( )
set ) )
/\ (dom g : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) V6() V35() V36() V37() ) PartFunc of ,) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ) : ( ( ) ( )
set ) ) & ex
r being ( ( ) (
V30()
real ext-real )
Real) st
(
r : ( ( ) (
V30()
real ext-real )
Real)
> 0 : ( ( ) (
empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V30()
real ext-real V71()
V72()
V73()
V74()
V75()
V76()
V77()
V93()
V94()
V101()
V104() )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V99()
V101() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
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() V101() V102() V104() ) set ) ) ,x0 : ( ( ) ( V30() real ext-real ) Real) .] : ( ( ) (
closed V71()
V72()
V73()
V104() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ) : ( ( ) ( )
set ) )
c= dom f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V6()
V35()
V36()
V37() )
PartFunc of ,) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
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() V101() V102() V104() ) set ) ) ,x0 : ( ( ) ( V30() real ext-real ) Real) .] : ( ( ) (
closed V71()
V72()
V73()
V104() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ) : ( ( ) ( )
set ) )
c= dom g : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V6()
V35()
V36()
V37() )
PartFunc of ,) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ) : ( ( ) ( )
set ) ) &
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V6()
V35()
V36()
V37() )
PartFunc of ,)
is_differentiable_on ].(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() V101() V102() V104() ) set ) ) ,x0 : ( ( ) ( V30() real ext-real ) Real) .[ : ( ( ) (
open V71()
V72()
V73()
V99()
V100()
V104() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ) : ( ( ) ( )
set ) ) &
g : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V6()
V35()
V36()
V37() )
PartFunc of ,)
is_differentiable_on ].(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() V101() V102() V104() ) set ) ) ,x0 : ( ( ) ( V30() real ext-real ) Real) .[ : ( ( ) (
open V71()
V72()
V73()
V99()
V100()
V104() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
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() V101() V102() V104() ) set ) ) ,x0 : ( ( ) ( V30() real ext-real ) Real) .[ : ( ( ) (
open V71()
V72()
V73()
V99()
V100()
V104() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ) : ( ( ) ( )
set ) )
c= dom (f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) V6() V35() V36() V37() ) PartFunc of ,) / g : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) V6() V35() V36() V37() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V6()
V35()
V36()
V37() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ) : ( ( ) (
V35()
V36()
V37() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
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() V101() V102() V104() ) set ) ) ,x0 : ( ( ) ( V30() real ext-real ) Real) .] : ( ( ) (
closed V71()
V72()
V73()
V104() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ) : ( ( ) ( )
set ) )
c= dom ((f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) V6() V35() V36() V37() ) 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() V101() V102() V104() ) set ) ) ,x0 : ( ( ) ( V30() real ext-real ) Real) .[ : ( ( ) ( open V71() V72() V73() V99() V100() V104() ) Element of K19(REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) V6() V35() V36() V37() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ,REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) : ( ( ) ( V35() V36() V37() ) set ) ) : ( ( ) ( ) set ) ) / (g : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) V6() V35() V36() V37() ) 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() V101() V102() V104() ) set ) ) ,x0 : ( ( ) ( V30() real ext-real ) Real) .[ : ( ( ) ( open V71() V72() V73() V99() V100() V104() ) Element of K19(REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) V6() V35() V36() V37() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ,REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) : ( ( ) ( V35() V36() V37() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V6()
V35()
V36()
V37() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ) : ( ( ) (
V35()
V36()
V37() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ) : ( ( ) ( )
set ) ) &
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V6()
V35()
V36()
V37() )
PartFunc of ,)
. x0 : ( ( ) (
V30()
real ext-real )
Real) : ( ( ) (
V30()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
= 0 : ( ( ) (
empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V30()
real ext-real V71()
V72()
V73()
V74()
V75()
V76()
V77()
V93()
V94()
V101()
V104() )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V99()
V101() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ) : ( ( ) ( )
set ) ) ) &
g : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V6()
V35()
V36()
V37() )
PartFunc of ,)
. x0 : ( ( ) (
V30()
real ext-real )
Real) : ( ( ) (
V30()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
= 0 : ( ( ) (
empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V30()
real ext-real V71()
V72()
V73()
V74()
V75()
V76()
V77()
V93()
V94()
V101()
V104() )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V99()
V101() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ) : ( ( ) ( )
set ) ) ) &
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V6()
V35()
V36()
V37() )
PartFunc of ,)
is_continuous_in x0 : ( ( ) (
V30()
real ext-real )
Real) &
g : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V6()
V35()
V36()
V37() )
PartFunc of ,)
is_continuous_in x0 : ( ( ) (
V30()
real ext-real )
Real) &
(f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) V6() V35() V36() V37() ) 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() V101() V102() V104() ) set ) ) ,x0 : ( ( ) ( V30() real ext-real ) Real) .[ : ( ( ) ( open V71() V72() V73() V99() V100() V104() ) Element of K19(REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V6()
V35()
V36()
V37() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ) : ( ( ) (
V35()
V36()
V37() )
set ) ) : ( ( ) ( )
set ) )
/ (g : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) V6() V35() V36() V37() ) 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() V101() V102() V104() ) set ) ) ,x0 : ( ( ) ( V30() real ext-real ) Real) .[ : ( ( ) ( open V71() V72() V73() V99() V100() V104() ) Element of K19(REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V6()
V35()
V36()
V37() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ) : ( ( ) (
V35()
V36()
V37() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V6()
V35()
V36()
V37() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ) : ( ( ) (
V35()
V36()
V37() )
set ) ) : ( ( ) ( )
set ) )
is_left_convergent_in x0 : ( ( ) (
V30()
real ext-real )
Real) ) holds
(
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V6()
V35()
V36()
V37() )
PartFunc of ,)
/ g : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V6()
V35()
V36()
V37() )
PartFunc of ,) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V6()
V35()
V36()
V37() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ) : ( ( ) (
V35()
V36()
V37() )
set ) ) : ( ( ) ( )
set ) )
is_left_convergent_in x0 : ( ( ) (
V30()
real ext-real )
Real) & ex
r being ( ( ) (
V30()
real ext-real )
Real) st
(
r : ( ( ) (
V30()
real ext-real )
Real)
> 0 : ( ( ) (
empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V30()
real ext-real V71()
V72()
V73()
V74()
V75()
V76()
V77()
V93()
V94()
V101()
V104() )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V99()
V101() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ) : ( ( ) ( )
set ) ) ) &
lim_left (
(f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) V6() V35() V36() V37() ) PartFunc of ,) / g : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) V6() V35() V36() V37() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V6()
V35()
V36()
V37() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ) : ( ( ) (
V35()
V36()
V37() )
set ) ) : ( ( ) ( )
set ) ) ,
x0 : ( ( ) (
V30()
real ext-real )
Real) ) : ( ( ) (
V30()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
= lim_left (
((f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) V6() V35() V36() V37() ) 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() V101() V102() V104() ) set ) ) ,x0 : ( ( ) ( V30() real ext-real ) Real) .[ : ( ( ) ( open V71() V72() V73() V99() V100() V104() ) Element of K19(REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) V6() V35() V36() V37() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ,REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) : ( ( ) ( V35() V36() V37() ) set ) ) : ( ( ) ( ) set ) ) / (g : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) V6() V35() V36() V37() ) 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() V101() V102() V104() ) set ) ) ,x0 : ( ( ) ( V30() real ext-real ) Real) .[ : ( ( ) ( open V71() V72() V73() V99() V100() V104() ) Element of K19(REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) V6() V35() V36() V37() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ,REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) : ( ( ) ( V35() V36() V37() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V6()
V35()
V36()
V37() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ) : ( ( ) (
V35()
V36()
V37() )
set ) ) : ( ( ) ( )
set ) ) ,
x0 : ( ( ) (
V30()
real ext-real )
Real) ) : ( ( ) (
V30()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ) ) ) ;
theorem
for
f,
g being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V6()
V35()
V36()
V37() )
PartFunc of ,)
for
x0 being ( ( ) (
V30()
real ext-real )
Real) st ex
N being ( ( ) (
open V71()
V72()
V73() )
Neighbourhood of
x0 : ( ( ) (
V30()
real ext-real )
Real) ) st
(
N : ( ( ) (
open V71()
V72()
V73() )
Neighbourhood of
b3 : ( ( ) (
V30()
real ext-real )
Real) )
c= dom f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V6()
V35()
V36()
V37() )
PartFunc of ,) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ) : ( ( ) ( )
set ) ) &
N : ( ( ) (
open V71()
V72()
V73() )
Neighbourhood of
b3 : ( ( ) (
V30()
real ext-real )
Real) )
c= dom g : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V6()
V35()
V36()
V37() )
PartFunc of ,) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ) : ( ( ) ( )
set ) ) &
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V6()
V35()
V36()
V37() )
PartFunc of ,)
is_differentiable_on N : ( ( ) (
open V71()
V72()
V73() )
Neighbourhood of
b3 : ( ( ) (
V30()
real ext-real )
Real) ) &
g : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V6()
V35()
V36()
V37() )
PartFunc of ,)
is_differentiable_on N : ( ( ) (
open V71()
V72()
V73() )
Neighbourhood of
b3 : ( ( ) (
V30()
real ext-real )
Real) ) &
N : ( ( ) (
open V71()
V72()
V73() )
Neighbourhood of
b3 : ( ( ) (
V30()
real ext-real )
Real) )
\ {x0 : ( ( ) ( V30() real ext-real ) Real) } : ( ( ) ( non
empty V71()
V72()
V73() )
set ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ) : ( ( ) ( )
set ) )
c= dom (f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) V6() V35() V36() V37() ) PartFunc of ,) / g : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) V6() V35() V36() V37() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V6()
V35()
V36()
V37() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ) : ( ( ) (
V35()
V36()
V37() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ) : ( ( ) ( )
set ) ) &
N : ( ( ) (
open V71()
V72()
V73() )
Neighbourhood of
b3 : ( ( ) (
V30()
real ext-real )
Real) )
c= dom ((f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) V6() V35() V36() V37() ) PartFunc of ,) `| N : ( ( ) ( open V71() V72() V73() ) Neighbourhood of b3 : ( ( ) ( V30() real ext-real ) Real) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) V6() V35() V36() V37() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ,REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) : ( ( ) ( V35() V36() V37() ) set ) ) : ( ( ) ( ) set ) ) / (g : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) V6() V35() V36() V37() ) PartFunc of ,) `| N : ( ( ) ( open V71() V72() V73() ) Neighbourhood of b3 : ( ( ) ( V30() real ext-real ) Real) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) V6() V35() V36() V37() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ,REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) : ( ( ) ( V35() V36() V37() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V6()
V35()
V36()
V37() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ) : ( ( ) (
V35()
V36()
V37() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ) : ( ( ) ( )
set ) ) &
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V6()
V35()
V36()
V37() )
PartFunc of ,)
. x0 : ( ( ) (
V30()
real ext-real )
Real) : ( ( ) (
V30()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
= 0 : ( ( ) (
empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V30()
real ext-real V71()
V72()
V73()
V74()
V75()
V76()
V77()
V93()
V94()
V101()
V104() )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V99()
V101() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ) : ( ( ) ( )
set ) ) ) &
g : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V6()
V35()
V36()
V37() )
PartFunc of ,)
. x0 : ( ( ) (
V30()
real ext-real )
Real) : ( ( ) (
V30()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
= 0 : ( ( ) (
empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V30()
real ext-real V71()
V72()
V73()
V74()
V75()
V76()
V77()
V93()
V94()
V101()
V104() )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V99()
V101() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ) : ( ( ) ( )
set ) ) ) &
(f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) V6() V35() V36() V37() ) PartFunc of ,) `| N : ( ( ) ( open V71() V72() V73() ) Neighbourhood of b3 : ( ( ) ( V30() real ext-real ) Real) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V6()
V35()
V36()
V37() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ) : ( ( ) (
V35()
V36()
V37() )
set ) ) : ( ( ) ( )
set ) )
/ (g : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) V6() V35() V36() V37() ) PartFunc of ,) `| N : ( ( ) ( open V71() V72() V73() ) Neighbourhood of b3 : ( ( ) ( V30() real ext-real ) Real) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V6()
V35()
V36()
V37() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ) : ( ( ) (
V35()
V36()
V37() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V6()
V35()
V36()
V37() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ) : ( ( ) (
V35()
V36()
V37() )
set ) ) : ( ( ) ( )
set ) )
is_convergent_in x0 : ( ( ) (
V30()
real ext-real )
Real) ) holds
(
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V6()
V35()
V36()
V37() )
PartFunc of ,)
/ g : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V6()
V35()
V36()
V37() )
PartFunc of ,) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V6()
V35()
V36()
V37() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ) : ( ( ) (
V35()
V36()
V37() )
set ) ) : ( ( ) ( )
set ) )
is_convergent_in x0 : ( ( ) (
V30()
real ext-real )
Real) & ex
N being ( ( ) (
open V71()
V72()
V73() )
Neighbourhood of
x0 : ( ( ) (
V30()
real ext-real )
Real) ) st
lim (
(f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) V6() V35() V36() V37() ) PartFunc of ,) / g : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) V6() V35() V36() V37() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V6()
V35()
V36()
V37() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ) : ( ( ) (
V35()
V36()
V37() )
set ) ) : ( ( ) ( )
set ) ) ,
x0 : ( ( ) (
V30()
real ext-real )
Real) ) : ( ( ) (
V30()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
= lim (
((f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) V6() V35() V36() V37() ) PartFunc of ,) `| N : ( ( ) ( open V71() V72() V73() ) Neighbourhood of b3 : ( ( ) ( V30() real ext-real ) Real) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) V6() V35() V36() V37() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ,REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) : ( ( ) ( V35() V36() V37() ) set ) ) : ( ( ) ( ) set ) ) / (g : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) V6() V35() V36() V37() ) PartFunc of ,) `| N : ( ( ) ( open V71() V72() V73() ) Neighbourhood of b3 : ( ( ) ( V30() real ext-real ) Real) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) V6() V35() V36() V37() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ,REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) : ( ( ) ( V35() V36() V37() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V6()
V35()
V36()
V37() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ) : ( ( ) (
V35()
V36()
V37() )
set ) ) : ( ( ) ( )
set ) ) ,
x0 : ( ( ) (
V30()
real ext-real )
Real) ) : ( ( ) (
V30()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ) ) ;
theorem
for
f,
g being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V6()
V35()
V36()
V37() )
PartFunc of ,)
for
x0 being ( ( ) (
V30()
real ext-real )
Real) st ex
N being ( ( ) (
open V71()
V72()
V73() )
Neighbourhood of
x0 : ( ( ) (
V30()
real ext-real )
Real) ) st
(
N : ( ( ) (
open V71()
V72()
V73() )
Neighbourhood of
b3 : ( ( ) (
V30()
real ext-real )
Real) )
c= dom f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V6()
V35()
V36()
V37() )
PartFunc of ,) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ) : ( ( ) ( )
set ) ) &
N : ( ( ) (
open V71()
V72()
V73() )
Neighbourhood of
b3 : ( ( ) (
V30()
real ext-real )
Real) )
c= dom g : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V6()
V35()
V36()
V37() )
PartFunc of ,) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ) : ( ( ) ( )
set ) ) &
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V6()
V35()
V36()
V37() )
PartFunc of ,)
is_differentiable_on N : ( ( ) (
open V71()
V72()
V73() )
Neighbourhood of
b3 : ( ( ) (
V30()
real ext-real )
Real) ) &
g : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V6()
V35()
V36()
V37() )
PartFunc of ,)
is_differentiable_on N : ( ( ) (
open V71()
V72()
V73() )
Neighbourhood of
b3 : ( ( ) (
V30()
real ext-real )
Real) ) &
N : ( ( ) (
open V71()
V72()
V73() )
Neighbourhood of
b3 : ( ( ) (
V30()
real ext-real )
Real) )
\ {x0 : ( ( ) ( V30() real ext-real ) Real) } : ( ( ) ( non
empty V71()
V72()
V73() )
set ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ) : ( ( ) ( )
set ) )
c= dom (f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) V6() V35() V36() V37() ) PartFunc of ,) / g : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) V6() V35() V36() V37() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V6()
V35()
V36()
V37() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ) : ( ( ) (
V35()
V36()
V37() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ) : ( ( ) ( )
set ) ) &
N : ( ( ) (
open V71()
V72()
V73() )
Neighbourhood of
b3 : ( ( ) (
V30()
real ext-real )
Real) )
c= dom ((f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) V6() V35() V36() V37() ) PartFunc of ,) `| N : ( ( ) ( open V71() V72() V73() ) Neighbourhood of b3 : ( ( ) ( V30() real ext-real ) Real) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) V6() V35() V36() V37() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ,REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) : ( ( ) ( V35() V36() V37() ) set ) ) : ( ( ) ( ) set ) ) / (g : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) V6() V35() V36() V37() ) PartFunc of ,) `| N : ( ( ) ( open V71() V72() V73() ) Neighbourhood of b3 : ( ( ) ( V30() real ext-real ) Real) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) V6() V35() V36() V37() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ,REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) : ( ( ) ( V35() V36() V37() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V6()
V35()
V36()
V37() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ) : ( ( ) (
V35()
V36()
V37() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V71()
V72()
V73() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ) : ( ( ) ( )
set ) ) &
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V6()
V35()
V36()
V37() )
PartFunc of ,)
. x0 : ( ( ) (
V30()
real ext-real )
Real) : ( ( ) (
V30()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
= 0 : ( ( ) (
empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V30()
real ext-real V71()
V72()
V73()
V74()
V75()
V76()
V77()
V93()
V94()
V101()
V104() )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V99()
V101() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ) : ( ( ) ( )
set ) ) ) &
g : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V6()
V35()
V36()
V37() )
PartFunc of ,)
. x0 : ( ( ) (
V30()
real ext-real )
Real) : ( ( ) (
V30()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
= 0 : ( ( ) (
empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V30()
real ext-real V71()
V72()
V73()
V74()
V75()
V76()
V77()
V93()
V94()
V101()
V104() )
Element of
NAT : ( ( ) ( non
empty epsilon-transitive epsilon-connected ordinal V71()
V72()
V73()
V74()
V75()
V76()
V77()
V99()
V101() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ) : ( ( ) ( )
set ) ) ) &
(f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) V6() V35() V36() V37() ) PartFunc of ,) `| N : ( ( ) ( open V71() V72() V73() ) Neighbourhood of b3 : ( ( ) ( V30() real ext-real ) Real) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V6()
V35()
V36()
V37() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ) : ( ( ) (
V35()
V36()
V37() )
set ) ) : ( ( ) ( )
set ) )
/ (g : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) V6() V35() V36() V37() ) PartFunc of ,) `| N : ( ( ) ( open V71() V72() V73() ) Neighbourhood of b3 : ( ( ) ( V30() real ext-real ) Real) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V6()
V35()
V36()
V37() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ) : ( ( ) (
V35()
V36()
V37() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V6()
V35()
V36()
V37() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ) : ( ( ) (
V35()
V36()
V37() )
set ) ) : ( ( ) ( )
set ) )
is_continuous_in x0 : ( ( ) (
V30()
real ext-real )
Real) ) holds
(
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V6()
V35()
V36()
V37() )
PartFunc of ,)
/ g : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V6()
V35()
V36()
V37() )
PartFunc of ,) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V6()
V35()
V36()
V37() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ) : ( ( ) (
V35()
V36()
V37() )
set ) ) : ( ( ) ( )
set ) )
is_convergent_in x0 : ( ( ) (
V30()
real ext-real )
Real) &
lim (
(f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) V6() V35() V36() V37() ) PartFunc of ,) / g : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) V6() V35() V36() V37() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
V6()
V35()
V36()
V37() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ) : ( ( ) (
V35()
V36()
V37() )
set ) ) : ( ( ) ( )
set ) ) ,
x0 : ( ( ) (
V30()
real ext-real )
Real) ) : ( ( ) (
V30()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
= (diff (f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) V6() V35() V36() V37() ) PartFunc of ,) ,x0 : ( ( ) ( V30() real ext-real ) Real) )) : ( ( ) (
V30()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) )
/ (diff (g : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V71() V72() V73() V77() V101() V102() V104() ) set ) ) V6() V35() V36() V37() ) PartFunc of ,) ,x0 : ( ( ) ( V30() real ext-real ) Real) )) : ( ( ) (
V30()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ) : ( ( ) (
V30()
real ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V71()
V72()
V73()
V77()
V101()
V102()
V104() )
set ) ) ) ;