:: ASYMPT_1 semantic presentation
begin
theorem
:: ASYMPT_1:1
for
t
,
t1
being ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
() )
Real_Sequence
) st
t
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
() )
Real_Sequence
)
.
0
: ( ( ) (
functional
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V25
()
real
ext-real
non
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
()
FinSequence-membered
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
=
0
: ( ( ) (
functional
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V25
()
real
ext-real
non
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
()
FinSequence-membered
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) & ( for
n
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) st
n
: ( (
Function-like
quasi_total
eventually-positive
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
)
>
0
: ( ( ) (
functional
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V25
()
real
ext-real
non
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
()
FinSequence-membered
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) holds
t
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
() )
Real_Sequence
)
.
n
: ( (
Function-like
quasi_total
eventually-positive
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
) : ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
=
(
(
(
(
12 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
*
(
n
: ( (
Function-like
quasi_total
eventually-positive
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
)
to_power
3 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
*
(
log
(2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ,
n
: ( (
Function-like
quasi_total
eventually-positive
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
) )
)
: ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
)
: ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
-
(
5 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
*
(
n
: ( (
Function-like
quasi_total
eventually-positive
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
)
^2
)
: ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
)
: ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
)
: ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
+
(
(
log
(2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ,
n
: ( (
Function-like
quasi_total
eventually-positive
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
) )
)
: ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
^2
)
: ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
)
: ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
+
36 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) ) & ( for
n
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) st
n
: ( (
Function-like
quasi_total
eventually-positive
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
)
>
0
: ( ( ) (
functional
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V25
()
real
ext-real
non
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
()
FinSequence-membered
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) holds
t1
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
() )
Real_Sequence
)
.
n
: ( (
Function-like
quasi_total
eventually-positive
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
) : ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
=
(
n
: ( (
Function-like
quasi_total
eventually-positive
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
)
to_power
3 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
*
(
log
(2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ,
n
: ( (
Function-like
quasi_total
eventually-positive
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
) )
)
: ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) ) holds
ex
s
,
s1
being ( (
Function-like
quasi_total
eventually-positive
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
) st
(
s
: ( (
Function-like
quasi_total
eventually-positive
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
)
=
t
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
() )
Real_Sequence
) &
s1
: ( (
Function-like
quasi_total
eventually-positive
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
)
=
t1
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
() )
Real_Sequence
) &
s
: ( (
Function-like
quasi_total
eventually-positive
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
)
in
Big_Oh
s1
: ( (
Function-like
quasi_total
eventually-positive
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
) : ( ( ) (
functional
non
empty
)
FUNCTION_DOMAIN
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) ) ;
theorem
:: ASYMPT_1:2
for
a
,
b
being ( (
logbase
) (
V25
()
real
ext-real
logbase
)
Real
)
for
f
,
g
being ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
() )
Real_Sequence
) st
a
: ( (
logbase
) (
V25
()
real
ext-real
logbase
)
Real
)
>
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) &
b
: ( (
logbase
) (
V25
()
real
ext-real
logbase
)
Real
)
>
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) & ( for
n
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) st
n
: ( (
Function-like
quasi_total
eventually-positive
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
)
>
0
: ( ( ) (
functional
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V25
()
real
ext-real
non
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
()
FinSequence-membered
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) holds
f
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
() )
Real_Sequence
)
.
n
: ( (
Function-like
quasi_total
eventually-positive
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
) : ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
=
log
(
a
: ( (
logbase
) (
V25
()
real
ext-real
logbase
)
Real
) ,
n
: ( (
Function-like
quasi_total
eventually-positive
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
) ) : ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) ) & ( for
n
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) st
n
: ( (
Function-like
quasi_total
eventually-positive
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
)
>
0
: ( ( ) (
functional
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V25
()
real
ext-real
non
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
()
FinSequence-membered
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) holds
g
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
() )
Real_Sequence
)
.
n
: ( (
Function-like
quasi_total
eventually-positive
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
) : ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
=
log
(
b
: ( (
logbase
) (
V25
()
real
ext-real
logbase
)
Real
) ,
n
: ( (
Function-like
quasi_total
eventually-positive
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
) ) : ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) ) holds
ex
s
,
s1
being ( (
Function-like
quasi_total
eventually-positive
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
) st
(
s
: ( (
Function-like
quasi_total
eventually-positive
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
)
=
f
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
() )
Real_Sequence
) &
s1
: ( (
Function-like
quasi_total
eventually-positive
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
)
=
g
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
() )
Real_Sequence
) &
Big_Oh
s
: ( (
Function-like
quasi_total
eventually-positive
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
) : ( ( ) (
functional
non
empty
)
FUNCTION_DOMAIN
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
=
Big_Oh
s1
: ( (
Function-like
quasi_total
eventually-positive
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
) : ( ( ) (
functional
non
empty
)
FUNCTION_DOMAIN
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) ) ;
definition
let
a
,
b
,
c
be ( ( ) (
V25
()
real
ext-real
)
Real
) ;
func
seq_a^
(
a
,
b
,
c
)
->
( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
() )
Real_Sequence
)
means
:: ASYMPT_1:def 1
for
n
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) holds
it
: ( ( ) ( )
Element
of
a
: ( ( ) (
functional
non
empty
)
FUNCTION_DOMAIN
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) )
.
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
=
a
: ( ( ) (
functional
non
empty
)
FUNCTION_DOMAIN
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
to_power
(
(
b
: ( ( ) (
functional
non
empty
)
FUNCTION_DOMAIN
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
*
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
+
c
: ( ( ) (
functional
non
empty
)
FUNCTION_DOMAIN
of
a
: ( ( ) (
functional
non
empty
)
FUNCTION_DOMAIN
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) ,
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
)
: ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) ;
end;
registration
let
a
be ( (
positive
) ( non
empty
V25
()
real
ext-real
positive
non
negative
)
Real
) ;
let
b
,
c
be ( ( ) (
V25
()
real
ext-real
)
Real
) ;
cluster
seq_a^
(
a
: ( (
positive
) ( non
empty
V25
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) ,
b
: ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) ,
c
: ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) ) : ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
() )
Real_Sequence
)
->
Function-like
quasi_total
eventually-positive
;
end;
theorem
:: ASYMPT_1:3
for
a
,
b
being ( (
positive
) ( non
empty
V25
()
real
ext-real
positive
non
negative
)
Real
) st
a
: ( (
positive
) ( non
empty
V25
()
real
ext-real
positive
non
negative
)
Real
)
<
b
: ( (
positive
) ( non
empty
V25
()
real
ext-real
positive
non
negative
)
Real
) holds
not
seq_a^
(
b
: ( (
positive
) ( non
empty
V25
()
real
ext-real
positive
non
negative
)
Real
) ,1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ,
0
: ( ( ) (
functional
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V25
()
real
ext-real
non
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
()
FinSequence-membered
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ) : ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
)
in
Big_Oh
(
seq_a^
(
a
: ( (
positive
) ( non
empty
V25
()
real
ext-real
positive
non
negative
)
Real
) ,1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ,
0
: ( ( ) (
functional
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V25
()
real
ext-real
non
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
()
FinSequence-membered
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) )
)
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
) : ( ( ) (
functional
non
empty
)
FUNCTION_DOMAIN
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) ;
definition
func
seq_logn
->
( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
() )
Real_Sequence
)
means
:: ASYMPT_1:def 2
(
it
: ( ( ) ( )
set
)
.
0
: ( ( ) (
functional
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V25
()
real
ext-real
non
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
()
FinSequence-membered
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
=
0
: ( ( ) (
functional
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V25
()
real
ext-real
non
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
()
FinSequence-membered
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) & ( for
n
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) st
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
>
0
: ( ( ) (
functional
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V25
()
real
ext-real
non
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
()
FinSequence-membered
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) holds
it
: ( ( ) ( )
set
)
.
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
=
log
(2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ,
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ) : ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) ) );
end;
definition
let
a
be ( ( ) (
V25
()
real
ext-real
)
Real
) ;
func
seq_n^
a
->
( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
() )
Real_Sequence
)
means
:: ASYMPT_1:def 3
(
it
: ( ( ) ( )
set
)
.
0
: ( ( ) (
functional
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V25
()
real
ext-real
non
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
()
FinSequence-membered
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
=
0
: ( ( ) (
functional
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V25
()
real
ext-real
non
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
()
FinSequence-membered
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) & ( for
n
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) st
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
>
0
: ( ( ) (
functional
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V25
()
real
ext-real
non
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
()
FinSequence-membered
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) holds
it
: ( ( ) ( )
set
)
.
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
=
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
to_power
a
: ( ( ) ( )
set
) : ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) ) );
end;
registration
cluster
seq_logn
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
() )
Real_Sequence
)
->
Function-like
quasi_total
eventually-positive
;
end;
registration
let
a
be ( ( ) (
V25
()
real
ext-real
)
Real
) ;
cluster
seq_n^
a
: ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
() )
Real_Sequence
)
->
Function-like
quasi_total
eventually-positive
;
end;
theorem
:: ASYMPT_1:4
for
f
,
g
being ( (
Function-like
quasi_total
eventually-nonnegative
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
)
Real_Sequence
) holds
(
Big_Oh
f
: ( (
Function-like
quasi_total
eventually-nonnegative
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
)
Real_Sequence
) : ( ( ) (
functional
non
empty
)
FUNCTION_DOMAIN
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
c=
Big_Oh
g
: ( (
Function-like
quasi_total
eventually-nonnegative
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
)
Real_Sequence
) : ( ( ) (
functional
non
empty
)
FUNCTION_DOMAIN
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) & not
Big_Oh
f
: ( (
Function-like
quasi_total
eventually-nonnegative
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
)
Real_Sequence
) : ( ( ) (
functional
non
empty
)
FUNCTION_DOMAIN
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
=
Big_Oh
g
: ( (
Function-like
quasi_total
eventually-nonnegative
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
)
Real_Sequence
) : ( ( ) (
functional
non
empty
)
FUNCTION_DOMAIN
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) iff (
f
: ( (
Function-like
quasi_total
eventually-nonnegative
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
)
Real_Sequence
)
in
Big_Oh
g
: ( (
Function-like
quasi_total
eventually-nonnegative
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
)
Real_Sequence
) : ( ( ) (
functional
non
empty
)
FUNCTION_DOMAIN
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) & not
f
: ( (
Function-like
quasi_total
eventually-nonnegative
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
)
Real_Sequence
)
in
Big_Omega
g
: ( (
Function-like
quasi_total
eventually-nonnegative
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
)
Real_Sequence
) : ( ( ) (
functional
non
empty
)
FUNCTION_DOMAIN
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) ) ) ;
theorem
:: ASYMPT_1:5
(
Big_Oh
seq_logn
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
) : ( ( ) (
functional
non
empty
)
FUNCTION_DOMAIN
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
c=
Big_Oh
(
seq_n^
(
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
/
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) (
V25
()
real
ext-real
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
)
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
) : ( ( ) (
functional
non
empty
)
FUNCTION_DOMAIN
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) & not
Big_Oh
seq_logn
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
) : ( ( ) (
functional
non
empty
)
FUNCTION_DOMAIN
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
=
Big_Oh
(
seq_n^
(
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
/
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) (
V25
()
real
ext-real
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
)
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
) : ( ( ) (
functional
non
empty
)
FUNCTION_DOMAIN
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) ) ;
theorem
:: ASYMPT_1:6
(
seq_n^
(
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
/
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) (
V25
()
real
ext-real
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
)
in
Big_Omega
seq_logn
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
) : ( ( ) (
functional
non
empty
)
FUNCTION_DOMAIN
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) & not
seq_logn
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
)
in
Big_Omega
(
seq_n^
(
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
/
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) (
V25
()
real
ext-real
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
)
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
) : ( ( ) (
functional
non
empty
)
FUNCTION_DOMAIN
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) ) ;
theorem
:: ASYMPT_1:7
for
f
being ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
() )
Real_Sequence
)
for
k
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) st ( for
n
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) holds
f
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
() )
Real_Sequence
)
.
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
=
Sum
(
(
seq_n^
k
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
) ,
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ) : ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) ) holds
f
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
() )
Real_Sequence
)
in
Big_Theta
(
seq_n^
(
k
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
+
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
) : ( ( ) (
functional
non
empty
)
FUNCTION_DOMAIN
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) ;
theorem
:: ASYMPT_1:8
for
f
being ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
() )
Real_Sequence
) st ( for
n
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) st
n
: ( (
Function-like
quasi_total
eventually-positive
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
)
>
0
: ( ( ) (
functional
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V25
()
real
ext-real
non
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
()
FinSequence-membered
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) holds
f
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
() )
Real_Sequence
)
.
n
: ( (
Function-like
quasi_total
eventually-positive
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
) : ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
=
n
: ( (
Function-like
quasi_total
eventually-positive
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
)
to_power
(
log
(2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ,
n
: ( (
Function-like
quasi_total
eventually-positive
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
) )
)
: ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) ) holds
ex
s
being ( (
Function-like
quasi_total
eventually-positive
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
) st
(
s
: ( (
Function-like
quasi_total
eventually-positive
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
)
=
f
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
() )
Real_Sequence
) & not
s
: ( (
Function-like
quasi_total
eventually-positive
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
) is
smooth
) ;
definition
let
b
be ( ( ) (
V25
()
real
ext-real
)
Real
) ;
func
seq_const
b
->
( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
() )
Real_Sequence
)
equals
:: ASYMPT_1:def 4
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) )
-->
b
: ( ( ) ( )
set
) : ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
T-Sequence-like
quasi_total
V35
()
V36
()
V37
() )
Element
of
K19
(
K20
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
V35
()
V36
()
V37
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ;
end;
registration
cluster
seq_const
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
() )
Real_Sequence
)
->
Function-like
quasi_total
eventually-nonnegative
;
end;
theorem
:: ASYMPT_1:9
for
f
being ( (
Function-like
quasi_total
eventually-nonnegative
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
)
Real_Sequence
) ex
F
being ( ( ) (
functional
non
empty
)
FUNCTION_DOMAIN
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) st
(
F
: ( ( ) (
functional
non
empty
)
FUNCTION_DOMAIN
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
=
{
(
seq_n^
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
)
}
: ( ( ) ( non
empty
)
set
) & (
f
: ( (
Function-like
quasi_total
eventually-nonnegative
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
)
Real_Sequence
)
in
F
: ( ( ) (
functional
non
empty
)
FUNCTION_DOMAIN
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
to_power
(
Big_Oh
(
seq_const
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
)
Real_Sequence
)
)
: ( ( ) (
functional
non
empty
)
FUNCTION_DOMAIN
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) (
functional
non
empty
)
FUNCTION_DOMAIN
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) implies ex
N
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ex
c
being ( ( ) (
V25
()
real
ext-real
)
Real
) ex
k
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) st
(
c
: ( ( ) (
V25
()
real
ext-real
)
Real
)
>
0
: ( ( ) (
functional
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V25
()
real
ext-real
non
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
()
FinSequence-membered
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) & ( for
n
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) st
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
>=
N
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) holds
( 1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
<=
f
: ( (
Function-like
quasi_total
eventually-nonnegative
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
)
Real_Sequence
)
.
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) &
f
: ( (
Function-like
quasi_total
eventually-nonnegative
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
)
Real_Sequence
)
.
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
<=
c
: ( ( ) (
V25
()
real
ext-real
)
Real
)
*
(
(
seq_n^
k
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
)
.
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) ) ) ) ) & ( ex
N
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ex
c
being ( ( ) (
V25
()
real
ext-real
)
Real
) ex
k
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) st
(
c
: ( ( ) (
V25
()
real
ext-real
)
Real
)
>
0
: ( ( ) (
functional
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V25
()
real
ext-real
non
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
()
FinSequence-membered
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) & ( for
n
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) st
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
>=
N
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) holds
( 1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
<=
f
: ( (
Function-like
quasi_total
eventually-nonnegative
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
)
Real_Sequence
)
.
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) &
f
: ( (
Function-like
quasi_total
eventually-nonnegative
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
)
Real_Sequence
)
.
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
<=
c
: ( ( ) (
V25
()
real
ext-real
)
Real
)
*
(
(
seq_n^
k
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
)
.
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) ) ) ) implies
f
: ( (
Function-like
quasi_total
eventually-nonnegative
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
)
Real_Sequence
)
in
F
: ( ( ) (
functional
non
empty
)
FUNCTION_DOMAIN
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
to_power
(
Big_Oh
(
seq_const
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
)
Real_Sequence
)
)
: ( ( ) (
functional
non
empty
)
FUNCTION_DOMAIN
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) (
functional
non
empty
)
FUNCTION_DOMAIN
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) ) ) ;
begin
theorem
:: ASYMPT_1:10
for
f
being ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
() )
Real_Sequence
) st ( for
n
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) holds
f
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
() )
Real_Sequence
)
.
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
=
(
(
3 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
*
(
10 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
to_power
6 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
-
(
(
18 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
*
(
10 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
to_power
3 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
*
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) (
V25
()
real
ext-real
integer
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
+
(
27 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
*
(
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
^2
)
: ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
)
: ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) ) holds
f
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
() )
Real_Sequence
)
in
Big_Oh
(
seq_n^
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
) : ( ( ) (
functional
non
empty
)
FUNCTION_DOMAIN
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) ;
begin
theorem
:: ASYMPT_1:11
seq_n^
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
)
in
Big_Oh
(
seq_n^
3 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
) : ( ( ) (
functional
non
empty
)
FUNCTION_DOMAIN
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) ;
theorem
:: ASYMPT_1:12
not
seq_n^
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
)
in
Big_Omega
(
seq_n^
3 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
) : ( ( ) (
functional
non
empty
)
FUNCTION_DOMAIN
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) ;
theorem
:: ASYMPT_1:13
ex
s
being ( (
Function-like
quasi_total
eventually-positive
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
) st
(
s
: ( (
Function-like
quasi_total
eventually-positive
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
)
=
seq_a^
(2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ,1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ,1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ) : ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
) &
seq_a^
(2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ,1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ,
0
: ( ( ) (
functional
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V25
()
real
ext-real
non
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
()
FinSequence-membered
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ) : ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
)
in
Big_Theta
s
: ( (
Function-like
quasi_total
eventually-positive
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
) : ( ( ) (
functional
non
empty
)
FUNCTION_DOMAIN
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) ) ;
definition
let
a
be ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ;
func
seq_n!
a
->
( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
() )
Real_Sequence
)
means
:: ASYMPT_1:def 5
for
n
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) holds
it
: ( ( ) ( )
set
)
.
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
=
(
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
+
a
: ( ( ) ( )
set
)
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
!
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ;
end;
registration
let
a
be ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ;
cluster
seq_n!
a
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
() )
Real_Sequence
)
->
Function-like
quasi_total
eventually-positive
;
end;
theorem
:: ASYMPT_1:14
not
seq_n!
0
: ( ( ) (
functional
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V25
()
real
ext-real
non
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
()
FinSequence-membered
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
)
in
Big_Theta
(
seq_n!
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
) : ( ( ) (
functional
non
empty
)
FUNCTION_DOMAIN
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) ;
begin
theorem
:: ASYMPT_1:15
for
f
being ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
() )
Real_Sequence
) st
f
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
() )
Real_Sequence
)
in
Big_Oh
(
seq_n^
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
) : ( ( ) (
functional
non
empty
)
FUNCTION_DOMAIN
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) holds
f
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
() )
Real_Sequence
)
(#)
f
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
() )
Real_Sequence
) : ( (
Function-like
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
() )
Element
of
K19
(
K20
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
V35
()
V36
()
V37
() )
set
) ) : ( ( ) ( non
empty
)
set
) )
in
Big_Oh
(
seq_n^
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
) : ( ( ) (
functional
non
empty
)
FUNCTION_DOMAIN
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) ;
begin
theorem
:: ASYMPT_1:16
ex
s
being ( (
Function-like
quasi_total
eventually-positive
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
) st
(
s
: ( (
Function-like
quasi_total
eventually-positive
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
)
=
seq_a^
(2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ,1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ,
0
: ( ( ) (
functional
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V25
()
real
ext-real
non
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
()
FinSequence-membered
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ) : ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
) & 2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
(#)
(
seq_n^
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
) : ( (
Function-like
quasi_total
eventually-nonnegative
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
)
Element
of
K19
(
K20
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
V35
()
V36
()
V37
() )
set
) ) : ( ( ) ( non
empty
)
set
) )
in
Big_Oh
(
seq_n^
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
) : ( ( ) (
functional
non
empty
)
FUNCTION_DOMAIN
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) & not
seq_a^
(2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ,2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ,
0
: ( ( ) (
functional
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V25
()
real
ext-real
non
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
()
FinSequence-membered
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ) : ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
)
in
Big_Oh
s
: ( (
Function-like
quasi_total
eventually-positive
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
) : ( ( ) (
functional
non
empty
)
FUNCTION_DOMAIN
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) ) ;
begin
theorem
:: ASYMPT_1:17
(
log
(2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ,3 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ) : ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
<
159 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
/
100 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) (
V25
()
real
ext-real
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) implies (
seq_n^
(
log
(2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ,3 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) )
)
: ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
)
in
Big_Oh
(
seq_n^
(
159 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
/
100 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) (
V25
()
real
ext-real
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
)
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
) : ( ( ) (
functional
non
empty
)
FUNCTION_DOMAIN
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) & not
seq_n^
(
log
(2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ,3 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) )
)
: ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
)
in
Big_Omega
(
seq_n^
(
159 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
/
100 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) (
V25
()
real
ext-real
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
)
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
) : ( ( ) (
functional
non
empty
)
FUNCTION_DOMAIN
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) & not
seq_n^
(
log
(2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ,3 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) )
)
: ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
)
in
Big_Theta
(
seq_n^
(
159 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
/
100 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) (
V25
()
real
ext-real
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
)
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
) : ( ( ) (
functional
non
empty
)
FUNCTION_DOMAIN
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) ) ) ;
begin
theorem
:: ASYMPT_1:18
for
f
,
g
being ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
() )
Real_Sequence
) st ( for
n
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) holds
f
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
() )
Real_Sequence
)
.
n
: ( (
Function-like
quasi_total
eventually-nonnegative
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
)
Real_Sequence
) : ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
=
n
: ( (
Function-like
quasi_total
eventually-nonnegative
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
)
Real_Sequence
)
mod
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( (
integer
) (
V25
()
real
ext-real
integer
)
set
) ) & ( for
n
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) holds
g
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
() )
Real_Sequence
)
.
n
: ( (
Function-like
quasi_total
eventually-nonnegative
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
)
Real_Sequence
) : ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
=
(
n
: ( (
Function-like
quasi_total
eventually-nonnegative
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
)
Real_Sequence
)
+
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
mod
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( (
integer
) (
V25
()
real
ext-real
integer
)
set
) ) holds
ex
s
,
s1
being ( (
Function-like
quasi_total
eventually-nonnegative
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
)
Real_Sequence
) st
(
s
: ( (
Function-like
quasi_total
eventually-nonnegative
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
)
Real_Sequence
)
=
f
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
() )
Real_Sequence
) &
s1
: ( (
Function-like
quasi_total
eventually-nonnegative
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
)
Real_Sequence
)
=
g
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
() )
Real_Sequence
) & not
s
: ( (
Function-like
quasi_total
eventually-nonnegative
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
)
Real_Sequence
)
in
Big_Oh
s1
: ( (
Function-like
quasi_total
eventually-nonnegative
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
)
Real_Sequence
) : ( ( ) (
functional
non
empty
)
FUNCTION_DOMAIN
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) & not
s1
: ( (
Function-like
quasi_total
eventually-nonnegative
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
)
Real_Sequence
)
in
Big_Oh
s
: ( (
Function-like
quasi_total
eventually-nonnegative
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
)
Real_Sequence
) : ( ( ) (
functional
non
empty
)
FUNCTION_DOMAIN
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) ) ;
begin
theorem
:: ASYMPT_1:19
for
f
,
g
being ( (
Function-like
quasi_total
eventually-nonnegative
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
)
Real_Sequence
) holds
(
Big_Oh
f
: ( (
Function-like
quasi_total
eventually-nonnegative
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
)
Real_Sequence
) : ( ( ) (
functional
non
empty
)
FUNCTION_DOMAIN
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
=
Big_Oh
g
: ( (
Function-like
quasi_total
eventually-nonnegative
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
)
Real_Sequence
) : ( ( ) (
functional
non
empty
)
FUNCTION_DOMAIN
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) iff
f
: ( (
Function-like
quasi_total
eventually-nonnegative
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
)
Real_Sequence
)
in
Big_Theta
g
: ( (
Function-like
quasi_total
eventually-nonnegative
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
)
Real_Sequence
) : ( ( ) (
functional
non
empty
)
FUNCTION_DOMAIN
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) ) ;
theorem
:: ASYMPT_1:20
for
f
,
g
being ( (
Function-like
quasi_total
eventually-nonnegative
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
)
Real_Sequence
) holds
(
f
: ( (
Function-like
quasi_total
eventually-nonnegative
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
)
Real_Sequence
)
in
Big_Theta
g
: ( (
Function-like
quasi_total
eventually-nonnegative
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
)
Real_Sequence
) : ( ( ) (
functional
non
empty
)
FUNCTION_DOMAIN
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) iff
Big_Theta
f
: ( (
Function-like
quasi_total
eventually-nonnegative
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
)
Real_Sequence
) : ( ( ) (
functional
non
empty
)
FUNCTION_DOMAIN
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
=
Big_Theta
g
: ( (
Function-like
quasi_total
eventually-nonnegative
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
)
Real_Sequence
) : ( ( ) (
functional
non
empty
)
FUNCTION_DOMAIN
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) ) ;
begin
theorem
:: ASYMPT_1:21
for
e
being ( ( ) (
V25
()
real
ext-real
)
Real
)
for
f
being ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
() )
Real_Sequence
) st
0
: ( ( ) (
functional
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V25
()
real
ext-real
non
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
()
FinSequence-membered
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
<
e
: ( ( ) (
V25
()
real
ext-real
)
Real
) & ( for
n
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) st
n
: ( (
Function-like
quasi_total
eventually-positive
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
)
>
0
: ( ( ) (
functional
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V25
()
real
ext-real
non
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
()
FinSequence-membered
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) holds
f
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
() )
Real_Sequence
)
.
n
: ( (
Function-like
quasi_total
eventually-positive
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
) : ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
=
n
: ( (
Function-like
quasi_total
eventually-positive
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
)
*
(
log
(2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ,
n
: ( (
Function-like
quasi_total
eventually-positive
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
) )
)
: ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) ) holds
ex
s
being ( (
Function-like
quasi_total
eventually-positive
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
) st
(
s
: ( (
Function-like
quasi_total
eventually-positive
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
)
=
f
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
() )
Real_Sequence
) &
Big_Oh
s
: ( (
Function-like
quasi_total
eventually-positive
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
) : ( ( ) (
functional
non
empty
)
FUNCTION_DOMAIN
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
c=
Big_Oh
(
seq_n^
(
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
+
e
: ( ( ) (
V25
()
real
ext-real
)
Real
)
)
: ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
)
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
) : ( ( ) (
functional
non
empty
)
FUNCTION_DOMAIN
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) & not
Big_Oh
s
: ( (
Function-like
quasi_total
eventually-positive
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
) : ( ( ) (
functional
non
empty
)
FUNCTION_DOMAIN
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
=
Big_Oh
(
seq_n^
(
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
+
e
: ( ( ) (
V25
()
real
ext-real
)
Real
)
)
: ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
)
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
) : ( ( ) (
functional
non
empty
)
FUNCTION_DOMAIN
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) ) ;
theorem
:: ASYMPT_1:22
for
e
being ( ( ) (
V25
()
real
ext-real
)
Real
)
for
g
being ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
() )
Real_Sequence
) st
e
: ( ( ) (
V25
()
real
ext-real
)
Real
)
<
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) & ( for
n
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) st
n
: ( (
Function-like
quasi_total
eventually-positive
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
)
>
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) holds
g
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
() )
Real_Sequence
)
.
n
: ( (
Function-like
quasi_total
eventually-positive
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
) : ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
=
(
n
: ( (
Function-like
quasi_total
eventually-positive
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
)
to_power
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
/
(
log
(2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ,
n
: ( (
Function-like
quasi_total
eventually-positive
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
) )
)
: ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) ) holds
ex
s
being ( (
Function-like
quasi_total
eventually-positive
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
) st
(
s
: ( (
Function-like
quasi_total
eventually-positive
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
)
=
g
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
() )
Real_Sequence
) &
Big_Oh
(
seq_n^
(
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
+
e
: ( ( ) (
V25
()
real
ext-real
)
Real
)
)
: ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
)
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
) : ( ( ) (
functional
non
empty
)
FUNCTION_DOMAIN
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
c=
Big_Oh
s
: ( (
Function-like
quasi_total
eventually-positive
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
) : ( ( ) (
functional
non
empty
)
FUNCTION_DOMAIN
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) & not
Big_Oh
(
seq_n^
(
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
+
e
: ( ( ) (
V25
()
real
ext-real
)
Real
)
)
: ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
)
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
) : ( ( ) (
functional
non
empty
)
FUNCTION_DOMAIN
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
=
Big_Oh
s
: ( (
Function-like
quasi_total
eventually-positive
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
) : ( ( ) (
functional
non
empty
)
FUNCTION_DOMAIN
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) ) ;
theorem
:: ASYMPT_1:23
for
f
being ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
() )
Real_Sequence
) st ( for
n
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) st
n
: ( (
Function-like
quasi_total
eventually-positive
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
)
>
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) holds
f
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
() )
Real_Sequence
)
.
n
: ( (
Function-like
quasi_total
eventually-positive
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
) : ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
=
(
n
: ( (
Function-like
quasi_total
eventually-positive
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
)
to_power
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
/
(
log
(2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ,
n
: ( (
Function-like
quasi_total
eventually-positive
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
) )
)
: ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) ) holds
ex
s
being ( (
Function-like
quasi_total
eventually-positive
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
) st
(
s
: ( (
Function-like
quasi_total
eventually-positive
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
)
=
f
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
() )
Real_Sequence
) &
Big_Oh
s
: ( (
Function-like
quasi_total
eventually-positive
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
) : ( ( ) (
functional
non
empty
)
FUNCTION_DOMAIN
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
c=
Big_Oh
(
seq_n^
8 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
) : ( ( ) (
functional
non
empty
)
FUNCTION_DOMAIN
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) & not
Big_Oh
s
: ( (
Function-like
quasi_total
eventually-positive
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
) : ( ( ) (
functional
non
empty
)
FUNCTION_DOMAIN
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
=
Big_Oh
(
seq_n^
8 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
) : ( ( ) (
functional
non
empty
)
FUNCTION_DOMAIN
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) ) ;
theorem
:: ASYMPT_1:24
for
g
being ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
() )
Real_Sequence
) st ( for
n
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) holds
g
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
() )
Real_Sequence
)
.
n
: ( (
Function-like
quasi_total
eventually-positive
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
) : ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
=
(
(
(
n
: ( (
Function-like
quasi_total
eventually-positive
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
)
^2
)
: ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
-
n
: ( (
Function-like
quasi_total
eventually-positive
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
)
)
: ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
+
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
to_power
4 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) ) holds
ex
s
being ( (
Function-like
quasi_total
eventually-positive
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
) st
(
s
: ( (
Function-like
quasi_total
eventually-positive
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
)
=
g
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
() )
Real_Sequence
) &
Big_Oh
(
seq_n^
8 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
) : ( ( ) (
functional
non
empty
)
FUNCTION_DOMAIN
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
=
Big_Oh
s
: ( (
Function-like
quasi_total
eventually-positive
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
) : ( ( ) (
functional
non
empty
)
FUNCTION_DOMAIN
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) ) ;
theorem
:: ASYMPT_1:25
for
e
being ( ( ) (
V25
()
real
ext-real
)
Real
) st
0
: ( ( ) (
functional
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V25
()
real
ext-real
non
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
()
FinSequence-membered
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
<
e
: ( ( ) (
V25
()
real
ext-real
)
Real
) &
e
: ( ( ) (
V25
()
real
ext-real
)
Real
)
<
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) holds
ex
s
being ( (
Function-like
quasi_total
eventually-positive
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
) st
(
s
: ( (
Function-like
quasi_total
eventually-positive
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
)
=
seq_a^
(
(
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
+
e
: ( ( ) (
V25
()
real
ext-real
)
Real
)
)
: ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) ,1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ,
0
: ( ( ) (
functional
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V25
()
real
ext-real
non
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
()
FinSequence-membered
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ) : ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
() )
Real_Sequence
) &
Big_Oh
(
seq_n^
8 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
) : ( ( ) (
functional
non
empty
)
FUNCTION_DOMAIN
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
c=
Big_Oh
s
: ( (
Function-like
quasi_total
eventually-positive
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
) : ( ( ) (
functional
non
empty
)
FUNCTION_DOMAIN
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) & not
Big_Oh
(
seq_n^
8 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
) : ( ( ) (
functional
non
empty
)
FUNCTION_DOMAIN
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
=
Big_Oh
s
: ( (
Function-like
quasi_total
eventually-positive
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
) : ( ( ) (
functional
non
empty
)
FUNCTION_DOMAIN
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) ) ;
begin
theorem
:: ASYMPT_1:26
for
f
,
g
being ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
() )
Real_Sequence
) st ( for
n
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) st
n
: ( (
Function-like
quasi_total
eventually-positive
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
)
>
0
: ( ( ) (
functional
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V25
()
real
ext-real
non
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
()
FinSequence-membered
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) holds
f
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
() )
Real_Sequence
)
.
n
: ( (
Function-like
quasi_total
eventually-positive
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
) : ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
=
n
: ( (
Function-like
quasi_total
eventually-positive
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
)
to_power
(
log
(2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ,
n
: ( (
Function-like
quasi_total
eventually-positive
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
) )
)
: ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) ) & ( for
n
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) st
n
: ( (
Function-like
quasi_total
eventually-positive
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
)
>
0
: ( ( ) (
functional
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V25
()
real
ext-real
non
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
()
FinSequence-membered
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) holds
g
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
() )
Real_Sequence
)
.
n
: ( (
Function-like
quasi_total
eventually-positive
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
) : ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
=
n
: ( (
Function-like
quasi_total
eventually-positive
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
)
to_power
(
sqrt
n
: ( (
Function-like
quasi_total
eventually-positive
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
)
)
: ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) ) holds
ex
s
,
s1
being ( (
Function-like
quasi_total
eventually-positive
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
) st
(
s
: ( (
Function-like
quasi_total
eventually-positive
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
)
=
f
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
() )
Real_Sequence
) &
s1
: ( (
Function-like
quasi_total
eventually-positive
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
)
=
g
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
() )
Real_Sequence
) &
Big_Oh
s
: ( (
Function-like
quasi_total
eventually-positive
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
) : ( ( ) (
functional
non
empty
)
FUNCTION_DOMAIN
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
c=
Big_Oh
s1
: ( (
Function-like
quasi_total
eventually-positive
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
) : ( ( ) (
functional
non
empty
)
FUNCTION_DOMAIN
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) & not
Big_Oh
s
: ( (
Function-like
quasi_total
eventually-positive
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
) : ( ( ) (
functional
non
empty
)
FUNCTION_DOMAIN
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
=
Big_Oh
s1
: ( (
Function-like
quasi_total
eventually-positive
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
) : ( ( ) (
functional
non
empty
)
FUNCTION_DOMAIN
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) ) ;
theorem
:: ASYMPT_1:27
for
f
being ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
() )
Real_Sequence
) st ( for
n
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) st
n
: ( (
Function-like
quasi_total
eventually-positive
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
)
>
0
: ( ( ) (
functional
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V25
()
real
ext-real
non
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
()
FinSequence-membered
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) holds
f
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
() )
Real_Sequence
)
.
n
: ( (
Function-like
quasi_total
eventually-positive
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
) : ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
=
n
: ( (
Function-like
quasi_total
eventually-positive
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
)
to_power
(
sqrt
n
: ( (
Function-like
quasi_total
eventually-positive
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
)
)
: ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) ) holds
ex
s
,
s1
being ( (
Function-like
quasi_total
eventually-positive
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
) st
(
s
: ( (
Function-like
quasi_total
eventually-positive
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
)
=
f
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
() )
Real_Sequence
) &
s1
: ( (
Function-like
quasi_total
eventually-positive
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
)
=
seq_a^
(2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ,1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ,
0
: ( ( ) (
functional
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V25
()
real
ext-real
non
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
()
FinSequence-membered
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ) : ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
) &
Big_Oh
s
: ( (
Function-like
quasi_total
eventually-positive
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
) : ( ( ) (
functional
non
empty
)
FUNCTION_DOMAIN
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
c=
Big_Oh
s1
: ( (
Function-like
quasi_total
eventually-positive
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
) : ( ( ) (
functional
non
empty
)
FUNCTION_DOMAIN
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) & not
Big_Oh
s
: ( (
Function-like
quasi_total
eventually-positive
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
) : ( ( ) (
functional
non
empty
)
FUNCTION_DOMAIN
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
=
Big_Oh
s1
: ( (
Function-like
quasi_total
eventually-positive
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
) : ( ( ) (
functional
non
empty
)
FUNCTION_DOMAIN
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) ) ;
theorem
:: ASYMPT_1:28
ex
s
,
s1
being ( (
Function-like
quasi_total
eventually-positive
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
) st
(
s
: ( (
Function-like
quasi_total
eventually-positive
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
)
=
seq_a^
(2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ,1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ,
0
: ( ( ) (
functional
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V25
()
real
ext-real
non
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
()
FinSequence-membered
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ) : ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
) &
s1
: ( (
Function-like
quasi_total
eventually-positive
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
)
=
seq_a^
(2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ,1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ,1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ) : ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
) &
Big_Oh
s
: ( (
Function-like
quasi_total
eventually-positive
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
) : ( ( ) (
functional
non
empty
)
FUNCTION_DOMAIN
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
=
Big_Oh
s1
: ( (
Function-like
quasi_total
eventually-positive
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
) : ( ( ) (
functional
non
empty
)
FUNCTION_DOMAIN
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) ) ;
theorem
:: ASYMPT_1:29
ex
s
,
s1
being ( (
Function-like
quasi_total
eventually-positive
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
) st
(
s
: ( (
Function-like
quasi_total
eventually-positive
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
)
=
seq_a^
(2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ,1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ,
0
: ( ( ) (
functional
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V25
()
real
ext-real
non
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
()
FinSequence-membered
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ) : ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
) &
s1
: ( (
Function-like
quasi_total
eventually-positive
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
)
=
seq_a^
(2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ,2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ,
0
: ( ( ) (
functional
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V25
()
real
ext-real
non
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
()
FinSequence-membered
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ) : ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
) &
Big_Oh
s
: ( (
Function-like
quasi_total
eventually-positive
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
) : ( ( ) (
functional
non
empty
)
FUNCTION_DOMAIN
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
c=
Big_Oh
s1
: ( (
Function-like
quasi_total
eventually-positive
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
) : ( ( ) (
functional
non
empty
)
FUNCTION_DOMAIN
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) & not
Big_Oh
s
: ( (
Function-like
quasi_total
eventually-positive
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
) : ( ( ) (
functional
non
empty
)
FUNCTION_DOMAIN
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
=
Big_Oh
s1
: ( (
Function-like
quasi_total
eventually-positive
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
) : ( ( ) (
functional
non
empty
)
FUNCTION_DOMAIN
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) ) ;
theorem
:: ASYMPT_1:30
ex
s
being ( (
Function-like
quasi_total
eventually-positive
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
) st
(
s
: ( (
Function-like
quasi_total
eventually-positive
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
)
=
seq_a^
(2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ,2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ,
0
: ( ( ) (
functional
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V25
()
real
ext-real
non
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
()
FinSequence-membered
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ) : ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
) &
Big_Oh
s
: ( (
Function-like
quasi_total
eventually-positive
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
) : ( ( ) (
functional
non
empty
)
FUNCTION_DOMAIN
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
c=
Big_Oh
(
seq_n!
0
: ( ( ) (
functional
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V25
()
real
ext-real
non
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
()
FinSequence-membered
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
) : ( ( ) (
functional
non
empty
)
FUNCTION_DOMAIN
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) & not
Big_Oh
s
: ( (
Function-like
quasi_total
eventually-positive
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
) : ( ( ) (
functional
non
empty
)
FUNCTION_DOMAIN
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
=
Big_Oh
(
seq_n!
0
: ( ( ) (
functional
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V25
()
real
ext-real
non
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
()
FinSequence-membered
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
) : ( ( ) (
functional
non
empty
)
FUNCTION_DOMAIN
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) ) ;
theorem
:: ASYMPT_1:31
(
Big_Oh
(
seq_n!
0
: ( ( ) (
functional
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V25
()
real
ext-real
non
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
()
FinSequence-membered
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
) : ( ( ) (
functional
non
empty
)
FUNCTION_DOMAIN
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
c=
Big_Oh
(
seq_n!
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
) : ( ( ) (
functional
non
empty
)
FUNCTION_DOMAIN
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) & not
Big_Oh
(
seq_n!
0
: ( ( ) (
functional
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V25
()
real
ext-real
non
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
()
FinSequence-membered
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
) : ( ( ) (
functional
non
empty
)
FUNCTION_DOMAIN
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
=
Big_Oh
(
seq_n!
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
) : ( ( ) (
functional
non
empty
)
FUNCTION_DOMAIN
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) ) ;
theorem
:: ASYMPT_1:32
for
g
being ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
() )
Real_Sequence
) st ( for
n
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) st
n
: ( (
Function-like
quasi_total
eventually-positive
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
)
>
0
: ( ( ) (
functional
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V25
()
real
ext-real
non
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
()
FinSequence-membered
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) holds
g
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
() )
Real_Sequence
)
.
n
: ( (
Function-like
quasi_total
eventually-positive
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
) : ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
=
n
: ( (
Function-like
quasi_total
eventually-positive
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
)
to_power
n
: ( (
Function-like
quasi_total
eventually-positive
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
) : ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ) holds
ex
s
being ( (
Function-like
quasi_total
eventually-positive
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
) st
(
s
: ( (
Function-like
quasi_total
eventually-positive
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
)
=
g
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
() )
Real_Sequence
) &
Big_Oh
(
seq_n!
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
) : ( ( ) (
functional
non
empty
)
FUNCTION_DOMAIN
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
c=
Big_Oh
s
: ( (
Function-like
quasi_total
eventually-positive
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
) : ( ( ) (
functional
non
empty
)
FUNCTION_DOMAIN
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) & not
Big_Oh
(
seq_n!
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
) : ( ( ) (
functional
non
empty
)
FUNCTION_DOMAIN
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
=
Big_Oh
s
: ( (
Function-like
quasi_total
eventually-positive
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
) : ( ( ) (
functional
non
empty
)
FUNCTION_DOMAIN
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) ) ;
begin
theorem
:: ASYMPT_1:33
for
n
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) st
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
>=
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) holds
for
f
being ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
() )
Real_Sequence
)
for
k
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) st ( for
n
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) holds
f
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
() )
Real_Sequence
)
.
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
=
Sum
(
(
seq_n^
k
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
) ,
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ) : ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) ) holds
f
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
() )
Real_Sequence
)
.
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
>=
(
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
to_power
(
k
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
+
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
/
(
k
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
+
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) (
V25
()
real
ext-real
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) ;
begin
theorem
:: ASYMPT_1:34
for
f
,
g
being ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
() )
Real_Sequence
) st ( for
n
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) st
n
: ( (
Function-like
quasi_total
eventually-nonnegative
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
)
Real_Sequence
)
>
0
: ( ( ) (
functional
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V25
()
real
ext-real
non
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
()
FinSequence-membered
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) holds
g
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
() )
Real_Sequence
)
.
n
: ( (
Function-like
quasi_total
eventually-nonnegative
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
)
Real_Sequence
) : ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
=
n
: ( (
Function-like
quasi_total
eventually-nonnegative
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
)
Real_Sequence
)
*
(
log
(2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ,
n
: ( (
Function-like
quasi_total
eventually-nonnegative
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
)
Real_Sequence
) )
)
: ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) ) & ( for
n
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) holds
f
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
() )
Real_Sequence
)
.
n
: ( (
Function-like
quasi_total
eventually-nonnegative
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
)
Real_Sequence
) : ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
=
log
(2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ,
(
n
: ( (
Function-like
quasi_total
eventually-nonnegative
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
)
Real_Sequence
)
!
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ) : ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) ) holds
ex
s
being ( (
Function-like
quasi_total
eventually-nonnegative
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
)
Real_Sequence
) st
(
s
: ( (
Function-like
quasi_total
eventually-nonnegative
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
)
Real_Sequence
)
=
g
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
() )
Real_Sequence
) &
f
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
() )
Real_Sequence
)
in
Big_Theta
s
: ( (
Function-like
quasi_total
eventually-nonnegative
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
)
Real_Sequence
) : ( ( ) (
functional
non
empty
)
FUNCTION_DOMAIN
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) ) ;
begin
theorem
:: ASYMPT_1:35
for
f
being ( (
Function-like
quasi_total
eventually-nonnegative
eventually-nondecreasing
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-nondecreasing
)
Real_Sequence
)
for
t
being ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
() )
Real_Sequence
) st ( for
n
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) holds
( (
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
mod
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( (
integer
) (
V25
()
real
ext-real
integer
)
set
)
=
0
: ( ( ) (
functional
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V25
()
real
ext-real
non
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
()
FinSequence-membered
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) implies
t
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
() )
Real_Sequence
)
.
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
=
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ) & (
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
mod
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( (
integer
) (
V25
()
real
ext-real
integer
)
set
)
=
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) implies
t
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
() )
Real_Sequence
)
.
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
=
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ) ) ) holds
not
t
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
() )
Real_Sequence
)
in
Big_Theta
f
: ( (
Function-like
quasi_total
eventually-nonnegative
eventually-nondecreasing
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-nondecreasing
)
Real_Sequence
) : ( ( ) (
functional
non
empty
)
FUNCTION_DOMAIN
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) ;
begin
begin
definition
func
POWEROF2SET
->
( ( non
empty
) ( non
empty
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Subset
of ( ( ) ( non
empty
)
set
) )
equals
:: ASYMPT_1:def 6
{
(
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
to_power
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) where
n
is ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : verum
}
;
end;
theorem
:: ASYMPT_1:36
for
f
being ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
() )
Real_Sequence
) st ( for
n
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) holds
( (
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
in
POWEROF2SET
: ( ( non
empty
) ( non
empty
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Subset
of ( ( ) ( non
empty
)
set
) ) implies
f
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
() )
Real_Sequence
)
.
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
=
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ) & ( not
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
in
POWEROF2SET
: ( ( non
empty
) ( non
empty
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Subset
of ( ( ) ( non
empty
)
set
) ) implies
f
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
() )
Real_Sequence
)
.
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
=
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
to_power
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ) ) ) holds
(
f
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
() )
Real_Sequence
)
in
Big_Theta
(
(
seq_n^
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
) ,
POWEROF2SET
: ( ( non
empty
) ( non
empty
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Subset
of ( ( ) ( non
empty
)
set
) ) ) : ( ( ) (
functional
non
empty
)
FUNCTION_DOMAIN
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) & not
f
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
() )
Real_Sequence
)
in
Big_Theta
(
seq_n^
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
) : ( ( ) (
functional
non
empty
)
FUNCTION_DOMAIN
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) &
seq_n^
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
) is
smooth
& not
f
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
() )
Real_Sequence
) is
eventually-nondecreasing
) ;
theorem
:: ASYMPT_1:37
for
f
,
g
being ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
() )
Real_Sequence
) st ( for
n
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) st
n
: ( (
Function-like
quasi_total
eventually-positive
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
)
>
0
: ( ( ) (
functional
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V25
()
real
ext-real
non
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
()
FinSequence-membered
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) holds
f
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
() )
Real_Sequence
)
.
n
: ( (
Function-like
quasi_total
eventually-positive
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
) : ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
=
n
: ( (
Function-like
quasi_total
eventually-positive
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
)
to_power
(
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
to_power
[\
(
log
(2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ,
n
: ( (
Function-like
quasi_total
eventually-positive
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
) )
)
: ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
/]
: ( (
integer
) (
V25
()
real
ext-real
integer
)
set
)
)
: ( (
real
) (
V25
()
real
ext-real
)
set
) : ( (
real
) (
V25
()
real
ext-real
)
set
) ) & ( for
n
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) st
n
: ( (
Function-like
quasi_total
eventually-positive
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
)
>
0
: ( ( ) (
functional
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V25
()
real
ext-real
non
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
()
FinSequence-membered
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) holds
g
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
() )
Real_Sequence
)
.
n
: ( (
Function-like
quasi_total
eventually-positive
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
) : ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
=
n
: ( (
Function-like
quasi_total
eventually-positive
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
)
to_power
n
: ( (
Function-like
quasi_total
eventually-positive
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
) : ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ) holds
ex
s
being ( (
Function-like
quasi_total
eventually-positive
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
) st
(
s
: ( (
Function-like
quasi_total
eventually-positive
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
)
=
g
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
() )
Real_Sequence
) &
f
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
() )
Real_Sequence
)
in
Big_Theta
(
s
: ( (
Function-like
quasi_total
eventually-positive
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
) ,
POWEROF2SET
: ( ( non
empty
) ( non
empty
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Subset
of ( ( ) ( non
empty
)
set
) ) ) : ( ( ) (
functional
non
empty
)
FUNCTION_DOMAIN
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) & not
f
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
() )
Real_Sequence
)
in
Big_Theta
s
: ( (
Function-like
quasi_total
eventually-positive
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
) : ( ( ) (
functional
non
empty
)
FUNCTION_DOMAIN
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) &
f
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
() )
Real_Sequence
) is
eventually-nondecreasing
&
s
: ( (
Function-like
quasi_total
eventually-positive
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
) is
eventually-nondecreasing
& not
s
: ( (
Function-like
quasi_total
eventually-positive
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
)
is_smooth_wrt
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ) ;
theorem
:: ASYMPT_1:38
for
g
being ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
() )
Real_Sequence
) st ( for
n
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) holds
( (
n
: ( (
Function-like
quasi_total
eventually-positive
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
)
in
POWEROF2SET
: ( ( non
empty
) ( non
empty
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Subset
of ( ( ) ( non
empty
)
set
) ) implies
g
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
() )
Real_Sequence
)
.
n
: ( (
Function-like
quasi_total
eventually-positive
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
) : ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
=
n
: ( (
Function-like
quasi_total
eventually-positive
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
) ) & ( not
n
: ( (
Function-like
quasi_total
eventually-positive
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
)
in
POWEROF2SET
: ( ( non
empty
) ( non
empty
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Subset
of ( ( ) ( non
empty
)
set
) ) implies
g
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
() )
Real_Sequence
)
.
n
: ( (
Function-like
quasi_total
eventually-positive
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
) : ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
=
n
: ( (
Function-like
quasi_total
eventually-positive
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
)
to_power
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ) ) ) holds
ex
s
being ( (
Function-like
quasi_total
eventually-positive
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
) st
(
s
: ( (
Function-like
quasi_total
eventually-positive
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
)
=
g
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
() )
Real_Sequence
) &
seq_n^
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
)
in
Big_Theta
(
s
: ( (
Function-like
quasi_total
eventually-positive
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
) ,
POWEROF2SET
: ( ( non
empty
) ( non
empty
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Subset
of ( ( ) ( non
empty
)
set
) ) ) : ( ( ) (
functional
non
empty
)
FUNCTION_DOMAIN
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) & not
seq_n^
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
)
in
Big_Theta
s
: ( (
Function-like
quasi_total
eventually-positive
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
) : ( ( ) (
functional
non
empty
)
FUNCTION_DOMAIN
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) &
s
: ( (
Function-like
quasi_total
eventually-positive
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
)
taken_every
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
() )
Element
of
K19
(
K20
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
V35
()
V36
()
V37
() )
set
) ) : ( ( ) ( non
empty
)
set
) )
in
Big_Oh
s
: ( (
Function-like
quasi_total
eventually-positive
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
) : ( ( ) (
functional
non
empty
)
FUNCTION_DOMAIN
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) &
seq_n^
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
) is
eventually-nondecreasing
& not
s
: ( (
Function-like
quasi_total
eventually-positive
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
) is
eventually-nondecreasing
) ;
begin
definition
let
x
be ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ;
func
Step1
x
->
( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
means
:: ASYMPT_1:def 7
ex
n
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) st
(
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
!
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
<=
x
: ( ( ) ( )
set
) &
x
: ( ( ) ( )
set
)
<
(
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
+
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
!
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) &
it
: ( ( ) ( )
set
)
=
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
!
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) )
if
x
: ( ( ) ( )
set
)
<>
0
: ( ( ) (
functional
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V25
()
real
ext-real
non
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
()
FinSequence-membered
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
otherwise
it
: ( ( ) ( )
set
)
=
0
: ( ( ) (
functional
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V25
()
real
ext-real
non
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
()
FinSequence-membered
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ;
end;
theorem
:: ASYMPT_1:39
for
f
being ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
() )
Real_Sequence
) st ( for
n
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) holds
f
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
() )
Real_Sequence
)
.
n
: ( (
Function-like
quasi_total
eventually-positive
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
) : ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
=
Step1
n
: ( (
Function-like
quasi_total
eventually-positive
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
) : ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ) holds
ex
s
being ( (
Function-like
quasi_total
eventually-positive
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
) st
(
s
: ( (
Function-like
quasi_total
eventually-positive
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
)
=
f
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
() )
Real_Sequence
) &
f
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
() )
Real_Sequence
) is
eventually-nondecreasing
& ( for
n
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) holds
f
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
() )
Real_Sequence
)
.
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
<=
(
seq_n^
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
)
.
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) ) & not
s
: ( (
Function-like
quasi_total
eventually-positive
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
) is
smooth
) ;
begin
theorem
:: ASYMPT_1:40
for
F
being ( (
Function-like
quasi_total
eventually-nonnegative
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
)
Real_Sequence
) st
F
: ( (
Function-like
quasi_total
eventually-nonnegative
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
)
Real_Sequence
)
=
(
seq_n^
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
)
-
(
seq_const
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
)
Real_Sequence
) : ( (
Function-like
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
() )
Element
of
K19
(
K20
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
V35
()
V36
()
V37
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) holds
(
Big_Theta
F
: ( (
Function-like
quasi_total
eventually-nonnegative
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
)
Real_Sequence
)
)
: ( ( ) (
functional
non
empty
)
FUNCTION_DOMAIN
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
+
(
Big_Theta
(
seq_n^
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
)
)
: ( ( ) (
functional
non
empty
)
FUNCTION_DOMAIN
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) (
functional
non
empty
)
FUNCTION_DOMAIN
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
=
Big_Theta
(
seq_n^
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
) : ( ( ) (
functional
non
empty
)
FUNCTION_DOMAIN
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) ;
begin
theorem
:: ASYMPT_1:41
ex
F
being ( ( ) (
functional
non
empty
)
FUNCTION_DOMAIN
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) st
(
F
: ( ( ) (
functional
non
empty
)
FUNCTION_DOMAIN
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
=
{
(
seq_n^
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
)
}
: ( ( ) ( non
empty
)
set
) & ( for
n
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) holds
(
seq_n^
(
-
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) (
V25
()
real
ext-real
non
positive
integer
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
)
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
)
.
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
<=
(
seq_n^
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
)
.
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) ) & not
seq_n^
(
-
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) (
V25
()
real
ext-real
non
positive
integer
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
)
in
F
: ( ( ) (
functional
non
empty
)
FUNCTION_DOMAIN
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
to_power
(
Big_Oh
(
seq_const
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
)
Real_Sequence
)
)
: ( ( ) (
functional
non
empty
)
FUNCTION_DOMAIN
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) (
functional
non
empty
)
FUNCTION_DOMAIN
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) ) ;
begin
theorem
:: ASYMPT_1:42
for
c
being ( ( non
negative
) (
V25
()
real
ext-real
non
negative
)
Real
)
for
x
,
f
being ( (
Function-like
quasi_total
eventually-nonnegative
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
)
Real_Sequence
) st ex
e
being ( ( ) (
V25
()
real
ext-real
)
Real
) ex
N
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) st
(
e
: ( ( ) (
V25
()
real
ext-real
)
Real
)
>
0
: ( ( ) (
functional
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V25
()
real
ext-real
non
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
()
FinSequence-membered
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) & ( for
n
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) st
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
>=
N
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) holds
f
: ( (
Function-like
quasi_total
eventually-nonnegative
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
)
Real_Sequence
)
.
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
>=
e
: ( ( ) (
V25
()
real
ext-real
)
Real
) ) ) &
x
: ( (
Function-like
quasi_total
eventually-nonnegative
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
)
Real_Sequence
)
in
Big_Oh
(
c
: ( ( non
negative
) (
V25
()
real
ext-real
non
negative
)
Real
)
+
f
: ( (
Function-like
quasi_total
eventually-nonnegative
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
)
Real_Sequence
)
)
: ( (
Function-like
quasi_total
eventually-nonnegative
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
)
Element
of
K19
(
K20
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
V35
()
V36
()
V37
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) : ( ( ) (
functional
non
empty
)
FUNCTION_DOMAIN
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) holds
x
: ( (
Function-like
quasi_total
eventually-nonnegative
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
)
Real_Sequence
)
in
Big_Oh
f
: ( (
Function-like
quasi_total
eventually-nonnegative
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
)
Real_Sequence
) : ( ( ) (
functional
non
empty
)
FUNCTION_DOMAIN
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) ;
begin
theorem
:: ASYMPT_1:43
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
to_power
12 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
=
4096 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ;
theorem
:: ASYMPT_1:44
for
n
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) st
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
>=
3 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) holds
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
^2
: ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
>
(
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
*
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
+
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ;
theorem
:: ASYMPT_1:45
for
n
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) st
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
>=
10 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) holds
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
to_power
(
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
-
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) (
V25
()
real
ext-real
integer
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
>
(
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
*
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
^2
: ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) ;
theorem
:: ASYMPT_1:46
for
n
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) st
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
>=
9 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) holds
(
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
+
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
to_power
6 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
<
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
*
(
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
to_power
6 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ;
theorem
:: ASYMPT_1:47
for
n
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) st
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
>=
30 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) holds
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
to_power
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
>
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
to_power
6 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ;
theorem
:: ASYMPT_1:48
for
x
being ( ( ) (
V25
()
real
ext-real
)
Real
) st
x
: ( ( ) (
V25
()
real
ext-real
)
Real
)
>
9 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) holds
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
to_power
x
: ( ( ) (
V25
()
real
ext-real
)
Real
) : ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
>
(
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
*
x
: ( ( ) (
V25
()
real
ext-real
)
Real
)
)
: ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
^2
: ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) ;
theorem
:: ASYMPT_1:49
ex
N
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) st
for
n
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) st
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
>=
N
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) holds
(
sqrt
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
-
(
log
(2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ,
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) )
)
: ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
>
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ;
theorem
:: ASYMPT_1:50
for
a
,
b
,
c
being ( ( ) (
V25
()
real
ext-real
)
Real
) st
a
: ( ( ) (
V25
()
real
ext-real
)
Real
)
>
0
: ( ( ) (
functional
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V25
()
real
ext-real
non
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
()
FinSequence-membered
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) &
c
: ( ( ) (
V25
()
real
ext-real
)
Real
)
>
0
: ( ( ) (
functional
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V25
()
real
ext-real
non
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
()
FinSequence-membered
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) &
c
: ( ( ) (
V25
()
real
ext-real
)
Real
)
<>
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) holds
a
: ( ( ) (
V25
()
real
ext-real
)
Real
)
to_power
b
: ( ( ) (
V25
()
real
ext-real
)
Real
) : ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
=
c
: ( ( ) (
V25
()
real
ext-real
)
Real
)
to_power
(
b
: ( ( ) (
V25
()
real
ext-real
)
Real
)
*
(
log
(
c
: ( ( ) (
V25
()
real
ext-real
)
Real
) ,
a
: ( ( ) (
V25
()
real
ext-real
)
Real
) )
)
: ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
)
: ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) ;
theorem
:: ASYMPT_1:51
5 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
!
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
=
120 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ;
theorem
:: ASYMPT_1:52
5 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
to_power
5 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
=
3125 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ;
theorem
:: ASYMPT_1:53
4 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
to_power
4 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
=
256 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ;
theorem
:: ASYMPT_1:54
for
n
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) holds
(
(
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
^2
)
: ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
-
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
+
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
>
0
: ( ( ) (
functional
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V25
()
real
ext-real
non
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
()
FinSequence-membered
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ;
theorem
:: ASYMPT_1:55
for
n
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) st
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
>=
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) holds
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
!
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
>
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ;
theorem
:: ASYMPT_1:56
for
n1
,
n
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) st
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
<=
n1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) holds
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
!
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
<=
n1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
!
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ;
theorem
:: ASYMPT_1:57
for
k
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) st
k
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
>=
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) holds
ex
n
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) st
(
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
!
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
<=
k
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) &
k
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
<
(
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
+
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
!
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) & ( for
m
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) st
m
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
!
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
<=
k
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) &
k
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
<
(
m
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
+
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
!
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) holds
m
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
=
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ) ) ;
theorem
:: ASYMPT_1:58
for
n
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) st
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
>=
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) holds
[/
(
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
/
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) (
V25
()
real
ext-real
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
\]
: ( (
integer
) (
V25
()
real
ext-real
integer
)
set
)
<
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ;
theorem
:: ASYMPT_1:59
for
n
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) st
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
>=
3 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) holds
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
!
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
>
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ;
theorem
:: ASYMPT_1:60
(
seq_n^
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
)
-
(
seq_const
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
)
Real_Sequence
) : ( (
Function-like
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
() )
Element
of
K19
(
K20
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
V35
()
V36
()
V37
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) is
eventually-positive
;
theorem
:: ASYMPT_1:61
for
n
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) st
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
>=
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) holds
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
to_power
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
>
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
+
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ;
theorem
:: ASYMPT_1:62
for
a
being ( (
logbase
) (
V25
()
real
ext-real
logbase
)
Real
)
for
f
being ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
() )
Real_Sequence
) st
a
: ( (
logbase
) (
V25
()
real
ext-real
logbase
)
Real
)
>
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) &
f
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
() )
Real_Sequence
)
.
0
: ( ( ) (
functional
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V25
()
real
ext-real
non
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
()
FinSequence-membered
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
=
0
: ( ( ) (
functional
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V25
()
real
ext-real
non
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
()
FinSequence-membered
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) & ( for
n
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) st
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
>
0
: ( ( ) (
functional
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V25
()
real
ext-real
non
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
()
FinSequence-membered
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) holds
f
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
() )
Real_Sequence
)
.
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
=
log
(
a
: ( (
logbase
) (
V25
()
real
ext-real
logbase
)
Real
) ,
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ) : ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) ) holds
f
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
() )
Real_Sequence
) is
eventually-positive
;
theorem
:: ASYMPT_1:63
for
f
,
g
being ( (
Function-like
quasi_total
eventually-nonnegative
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
)
Real_Sequence
) holds
( (
f
: ( (
Function-like
quasi_total
eventually-nonnegative
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
)
Real_Sequence
)
in
Big_Oh
g
: ( (
Function-like
quasi_total
eventually-nonnegative
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
)
Real_Sequence
) : ( ( ) (
functional
non
empty
)
FUNCTION_DOMAIN
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) &
g
: ( (
Function-like
quasi_total
eventually-nonnegative
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
)
Real_Sequence
)
in
Big_Oh
f
: ( (
Function-like
quasi_total
eventually-nonnegative
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
)
Real_Sequence
) : ( ( ) (
functional
non
empty
)
FUNCTION_DOMAIN
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) ) iff
Big_Oh
f
: ( (
Function-like
quasi_total
eventually-nonnegative
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
)
Real_Sequence
) : ( ( ) (
functional
non
empty
)
FUNCTION_DOMAIN
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
=
Big_Oh
g
: ( (
Function-like
quasi_total
eventually-nonnegative
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
)
Real_Sequence
) : ( ( ) (
functional
non
empty
)
FUNCTION_DOMAIN
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) ) ;
theorem
:: ASYMPT_1:64
for
a
,
b
,
c
being ( ( ) (
V25
()
real
ext-real
)
Real
) st
0
: ( ( ) (
functional
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V25
()
real
ext-real
non
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
()
FinSequence-membered
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
<
a
: ( ( ) (
V25
()
real
ext-real
)
Real
) &
a
: ( ( ) (
V25
()
real
ext-real
)
Real
)
<=
b
: ( ( ) (
V25
()
real
ext-real
)
Real
) &
c
: ( ( ) (
V25
()
real
ext-real
)
Real
)
>=
0
: ( ( ) (
functional
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V25
()
real
ext-real
non
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
()
FinSequence-membered
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) holds
a
: ( ( ) (
V25
()
real
ext-real
)
Real
)
to_power
c
: ( ( ) (
V25
()
real
ext-real
)
Real
) : ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
<=
b
: ( ( ) (
V25
()
real
ext-real
)
Real
)
to_power
c
: ( ( ) (
V25
()
real
ext-real
)
Real
) : ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) ;
theorem
:: ASYMPT_1:65
for
n
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) st
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
>=
4 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) holds
(
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
*
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
+
3 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
<
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
to_power
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ;
theorem
:: ASYMPT_1:66
for
n
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) st
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
>=
6 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) holds
(
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
+
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
^2
: ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
<
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
to_power
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ;
theorem
:: ASYMPT_1:67
for
c
being ( ( ) (
V25
()
real
ext-real
)
Real
) st
c
: ( ( ) (
V25
()
real
ext-real
)
Real
)
>
6 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) holds
c
: ( ( ) (
V25
()
real
ext-real
)
Real
)
^2
: ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
<
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
to_power
c
: ( ( ) (
V25
()
real
ext-real
)
Real
) : ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) ;
theorem
:: ASYMPT_1:68
for
e
being ( (
positive
) ( non
empty
V25
()
real
ext-real
positive
non
negative
)
Real
)
for
f
being ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
() )
Real_Sequence
) st
f
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
() )
Real_Sequence
)
.
0
: ( ( ) (
functional
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V25
()
real
ext-real
non
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
()
FinSequence-membered
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
=
0
: ( ( ) (
functional
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V25
()
real
ext-real
non
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
()
FinSequence-membered
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) & ( for
n
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) st
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
>
0
: ( ( ) (
functional
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V25
()
real
ext-real
non
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
()
FinSequence-membered
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) holds
f
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
() )
Real_Sequence
)
.
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
=
log
(2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ,
(
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
to_power
e
: ( (
positive
) ( non
empty
V25
()
real
ext-real
positive
non
negative
)
Real
)
)
: ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) ) : ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) ) holds
(
f
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
() )
Real_Sequence
)
/"
(
seq_n^
e
: ( (
positive
) ( non
empty
V25
()
real
ext-real
positive
non
negative
)
Real
)
)
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
) : ( (
Function-like
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
() )
Element
of
K19
(
K20
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
V35
()
V36
()
V37
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) is
convergent
&
lim
(
f
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
() )
Real_Sequence
)
/"
(
seq_n^
e
: ( (
positive
) ( non
empty
V25
()
real
ext-real
positive
non
negative
)
Real
)
)
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
)
)
: ( (
Function-like
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
() )
Element
of
K19
(
K20
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
V35
()
V36
()
V37
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) : ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
=
0
: ( ( ) (
functional
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V25
()
real
ext-real
non
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
()
FinSequence-membered
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ) ;
theorem
:: ASYMPT_1:69
for
e
being ( ( ) (
V25
()
real
ext-real
)
Real
) st
e
: ( ( ) (
V25
()
real
ext-real
)
Real
)
>
0
: ( ( ) (
functional
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V25
()
real
ext-real
non
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
()
FinSequence-membered
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) holds
(
seq_logn
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
)
/"
(
seq_n^
e
: ( ( ) (
V25
()
real
ext-real
)
Real
)
)
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
) : ( (
Function-like
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
() )
Element
of
K19
(
K20
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
V35
()
V36
()
V37
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) is
convergent
&
lim
(
seq_logn
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
)
/"
(
seq_n^
e
: ( ( ) (
V25
()
real
ext-real
)
Real
)
)
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
)
)
: ( (
Function-like
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
() )
Element
of
K19
(
K20
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
V35
()
V36
()
V37
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) : ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
=
0
: ( ( ) (
functional
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V25
()
real
ext-real
non
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
()
FinSequence-membered
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ) ;
theorem
:: ASYMPT_1:70
for
f
being ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
() )
Real_Sequence
)
for
N
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) st ( for
n
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) st
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
<=
N
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) holds
f
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
() )
Real_Sequence
)
.
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
>=
0
: ( ( ) (
functional
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V25
()
real
ext-real
non
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
()
FinSequence-membered
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ) holds
Sum
(
f
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
() )
Real_Sequence
) ,
N
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ) : ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
>=
0
: ( ( ) (
functional
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V25
()
real
ext-real
non
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
()
FinSequence-membered
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ;
theorem
:: ASYMPT_1:71
for
f
,
g
being ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
() )
Real_Sequence
)
for
N
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) st ( for
n
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) st
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
<=
N
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) holds
f
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
() )
Real_Sequence
)
.
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
<=
g
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
() )
Real_Sequence
)
.
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) ) holds
Sum
(
f
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
() )
Real_Sequence
) ,
N
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ) : ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
<=
Sum
(
g
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
() )
Real_Sequence
) ,
N
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ) : ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) ;
theorem
:: ASYMPT_1:72
for
f
being ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
() )
Real_Sequence
)
for
b
being ( ( ) (
V25
()
real
ext-real
)
Real
) st
f
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
() )
Real_Sequence
)
.
0
: ( ( ) (
functional
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V25
()
real
ext-real
non
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
()
FinSequence-membered
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
=
0
: ( ( ) (
functional
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V25
()
real
ext-real
non
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
()
FinSequence-membered
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) & ( for
n
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) st
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
>
0
: ( ( ) (
functional
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V25
()
real
ext-real
non
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
()
FinSequence-membered
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) holds
f
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
() )
Real_Sequence
)
.
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
=
b
: ( ( ) (
V25
()
real
ext-real
)
Real
) ) holds
for
N
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) holds
Sum
(
f
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
() )
Real_Sequence
) ,
N
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ) : ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
=
b
: ( ( ) (
V25
()
real
ext-real
)
Real
)
*
N
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) ;
theorem
:: ASYMPT_1:73
for
f
being ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
() )
Real_Sequence
)
for
N
,
M
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) holds
(
Sum
(
f
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
() )
Real_Sequence
) ,
N
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ,
M
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) )
)
: ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
+
(
f
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
() )
Real_Sequence
)
.
(
N
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
+
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
=
Sum
(
f
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
() )
Real_Sequence
) ,
(
N
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
+
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ,
M
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ) : ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) ;
theorem
:: ASYMPT_1:74
for
f
,
g
being ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
() )
Real_Sequence
)
for
M
,
N
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) st
N
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
>=
M
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
+
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) & ( for
n
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) st
M
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
+
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
<=
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) &
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
<=
N
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) holds
f
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
() )
Real_Sequence
)
.
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
<=
g
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
() )
Real_Sequence
)
.
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) ) holds
Sum
(
f
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
() )
Real_Sequence
) ,
N
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ,
M
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ) : ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
<=
Sum
(
g
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
() )
Real_Sequence
) ,
N
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ,
M
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ) : ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) ;
theorem
:: ASYMPT_1:75
for
n
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) holds
[/
(
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
/
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) (
V25
()
real
ext-real
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
\]
: ( (
integer
) (
V25
()
real
ext-real
integer
)
set
)
<=
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ;
theorem
:: ASYMPT_1:76
for
f
being ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
() )
Real_Sequence
)
for
b
being ( ( ) (
V25
()
real
ext-real
)
Real
)
for
N
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) st
f
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
() )
Real_Sequence
)
.
0
: ( ( ) (
functional
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V25
()
real
ext-real
non
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
()
FinSequence-membered
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
=
0
: ( ( ) (
functional
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V25
()
real
ext-real
non
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
()
FinSequence-membered
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) & ( for
n
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) st
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
>
0
: ( ( ) (
functional
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V25
()
real
ext-real
non
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
()
FinSequence-membered
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) holds
f
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
() )
Real_Sequence
)
.
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
=
b
: ( ( ) (
V25
()
real
ext-real
)
Real
) ) holds
for
M
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) holds
Sum
(
f
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
() )
Real_Sequence
) ,
N
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ,
M
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ) : ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
=
b
: ( ( ) (
V25
()
real
ext-real
)
Real
)
*
(
N
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
-
M
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) (
V25
()
real
ext-real
integer
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) ;
theorem
:: ASYMPT_1:77
for
f
,
g
being ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
() )
Real_Sequence
)
for
N
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
for
c
being ( ( ) (
V25
()
real
ext-real
)
Real
) st
f
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
() )
Real_Sequence
) is
convergent
&
lim
f
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
() )
Real_Sequence
) : ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
=
c
: ( ( ) (
V25
()
real
ext-real
)
Real
) & ( for
n
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) st
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
>=
N
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) holds
f
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
() )
Real_Sequence
)
.
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
=
g
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
() )
Real_Sequence
)
.
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) ) holds
(
g
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
() )
Real_Sequence
) is
convergent
&
lim
g
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
() )
Real_Sequence
) : ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
=
c
: ( ( ) (
V25
()
real
ext-real
)
Real
) ) ;
theorem
:: ASYMPT_1:78
for
n
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) st
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
>=
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) holds
(
(
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
^2
)
: ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
-
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
+
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
<=
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
^2
: ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) ;
theorem
:: ASYMPT_1:79
for
n
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) st
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
>=
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) holds
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
^2
: ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
<=
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
*
(
(
(
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
^2
)
: ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
-
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
+
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) ;
theorem
:: ASYMPT_1:80
for
e
being ( ( ) (
V25
()
real
ext-real
)
Real
) st
0
: ( ( ) (
functional
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V25
()
real
ext-real
non
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
()
FinSequence-membered
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
<
e
: ( ( ) (
V25
()
real
ext-real
)
Real
) &
e
: ( ( ) (
V25
()
real
ext-real
)
Real
)
<
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) holds
ex
N
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) st
for
n
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) st
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
>=
N
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) holds
(
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
*
(
log
(2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ,
(
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
+
e
: ( ( ) (
V25
()
real
ext-real
)
Real
)
)
: ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) )
)
: ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
)
: ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
-
(
8 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
*
(
log
(2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ,
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) )
)
: ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
)
: ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
>
8 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
*
(
log
(2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ,
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) )
)
: ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) ;
theorem
:: ASYMPT_1:81
for
n
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) st
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
>=
10 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) holds
(
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
to_power
(
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
*
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
/
(
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
!
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) (
V25
()
real
ext-real
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
<
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
/
(
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
to_power
(
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
-
9 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) (
V25
()
real
ext-real
integer
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
)
: ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) ;
theorem
:: ASYMPT_1:82
for
n
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) st
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
>=
3 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) holds
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
*
(
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
-
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) (
V25
()
real
ext-real
integer
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) (
V25
()
real
ext-real
integer
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
>=
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
-
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) (
V25
()
real
ext-real
integer
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) ;
theorem
:: ASYMPT_1:83
for
c
being ( (
real
) (
V25
()
real
ext-real
)
number
) st
c
: ( (
real
) (
V25
()
real
ext-real
)
number
)
>=
0
: ( ( ) (
functional
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V25
()
real
ext-real
non
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
()
FinSequence-membered
)
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) holds
c
: ( (
real
) (
V25
()
real
ext-real
)
number
)
to_power
(
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
/
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) (
V25
()
real
ext-real
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( (
real
) (
V25
()
real
ext-real
)
set
)
=
sqrt
c
: ( (
real
) (
V25
()
real
ext-real
)
number
) : ( (
real
) (
V25
()
real
ext-real
)
set
) ;
theorem
:: ASYMPT_1:84
ex
N
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) st
for
n
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) st
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
>=
N
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) holds
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
-
(
(
sqrt
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
*
(
log
(2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ,
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) )
)
: ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
)
: ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
>
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
/
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) (
V25
()
real
ext-real
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) ;
theorem
:: ASYMPT_1:85
for
s
being ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
() )
Real_Sequence
) st ( for
n
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) holds
s
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
() )
Real_Sequence
)
.
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
=
(
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
+
(
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
/
(
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
+
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) (
V25
()
real
ext-real
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
)
: ( ( ) ( non
empty
V25
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
to_power
(
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
+
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) ) holds
s
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
() )
Real_Sequence
) is
V41
() ;
theorem
:: ASYMPT_1:86
for
n
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) st
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
>=
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) holds
(
(
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
+
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
/
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) (
V25
()
real
ext-real
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
to_power
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
<=
(
(
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
+
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
/
(
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
+
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) (
V25
()
real
ext-real
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
to_power
(
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
+
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) ;
theorem
:: ASYMPT_1:87
for
k
,
n
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) st
k
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
<=
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) holds
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
choose
k
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
>=
(
(
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
+
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
choose
k
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
/
(
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
+
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) (
V25
()
real
ext-real
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) ;
theorem
:: ASYMPT_1:88
for
f
being ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
() )
Real_Sequence
) st ( for
n
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) holds
f
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
() )
Real_Sequence
)
.
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
=
log
(2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ,
(
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
!
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ) : ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) ) holds
for
n
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) holds
f
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
() )
Real_Sequence
)
.
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
=
Sum
(
seq_logn
: ( (
Function-like
quasi_total
) (
V1
()
V4
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
V5
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
Function-like
non
empty
V14
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
quasi_total
V35
()
V36
()
V37
()
eventually-nonnegative
eventually-positive
eventually-nonzero
)
Real_Sequence
) ,
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ) : ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) ;
theorem
:: ASYMPT_1:89
for
n
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) st
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
>=
4 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) holds
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
*
(
log
(2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ,
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) )
)
: ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
>=
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
*
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ;
theorem
:: ASYMPT_1:90
for
n
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) st
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
>=
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) holds
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
^2
: ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
>
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
+
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ;
theorem
:: ASYMPT_1:91
for
n
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) st
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
>=
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) holds
(
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
to_power
(
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
+
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
-
(
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
to_power
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) (
V25
()
real
ext-real
integer
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
>
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ;
theorem
:: ASYMPT_1:92
for
n
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) st
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
>=
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) holds
not
(
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
to_power
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
-
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) (
V25
()
real
ext-real
integer
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
in
POWEROF2SET
: ( ( non
empty
) ( non
empty
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Subset
of ( ( ) ( non
empty
)
set
) ) ;
theorem
:: ASYMPT_1:93
for
n
,
k
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) st
k
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
>=
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) &
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
!
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
<=
k
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) &
k
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
<
(
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
+
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
!
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) holds
Step1
k
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
=
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
!
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ;
theorem
:: ASYMPT_1:94
for
a
,
b
,
c
being ( ( ) (
V25
()
real
ext-real
)
Real
) st
a
: ( ( ) (
V25
()
real
ext-real
)
Real
)
>
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) &
b
: ( ( ) (
V25
()
real
ext-real
)
Real
)
>=
a
: ( ( ) (
V25
()
real
ext-real
)
Real
) &
c
: ( ( ) (
V25
()
real
ext-real
)
Real
)
>=
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V25
()
real
ext-real
positive
non
negative
integer
V48
()
V49
()
V50
()
V51
()
V52
()
V53
()
V54
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V49
()
V50
()
V51
()
V52
()
V53
()
V54
()
V55
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) holds
log
(
a
: ( ( ) (
V25
()
real
ext-real
)
Real
) ,
c
: ( ( ) (
V25
()
real
ext-real
)
Real
) ) : ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) )
>=
log
(
b
: ( ( ) (
V25
()
real
ext-real
)
Real
) ,
c
: ( ( ) (
V25
()
real
ext-real
)
Real
) ) : ( ( ) (
V25
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V49
()
V50
()
V51
()
V55
()
V56
() )
set
) ) ;