:: ASYMPT_1 semantic presentation

REAL is non empty V49() V50() V51() V55() V56() set
NAT is non empty epsilon-transitive epsilon-connected ordinal V49() V50() V51() V52() V53() V54() V55() Element of K19(REAL)
K19(REAL) is non empty set
COMPLEX is non empty V49() V55() V56() set
RAT is non empty V49() V50() V51() V52() V55() V56() set
INT is non empty V49() V50() V51() V52() V53() V55() V56() set
omega is non empty epsilon-transitive epsilon-connected ordinal V49() V50() V51() V52() V53() V54() V55() set
K19(omega) is non empty set
K19(NAT) is non empty set
K20(REAL,REAL) is non empty V35() V36() V37() set
K19(K20(REAL,REAL)) is non empty set
K20(NAT,REAL) is non empty V35() V36() V37() set
K19(K20(NAT,REAL)) is non empty set
K20(COMPLEX,COMPLEX) is non empty V35() set
K19(K20(COMPLEX,COMPLEX)) is non empty set
K20(K20(COMPLEX,COMPLEX),COMPLEX) is non empty V35() set
K19(K20(K20(COMPLEX,COMPLEX),COMPLEX)) is non empty set
K20(K20(REAL,REAL),REAL) is non empty V35() V36() V37() set
K19(K20(K20(REAL,REAL),REAL)) is non empty set
K20(RAT,RAT) is V5( RAT ) non empty V35() V36() V37() set
K19(K20(RAT,RAT)) is non empty set
K20(K20(RAT,RAT),RAT) is V5( RAT ) non empty V35() V36() V37() set
K19(K20(K20(RAT,RAT),RAT)) is non empty set
K20(INT,INT) is V5( RAT ) V5( INT ) non empty V35() V36() V37() set
K19(K20(INT,INT)) is non empty set
K20(K20(INT,INT),INT) is V5( RAT ) V5( INT ) non empty V35() V36() V37() set
K19(K20(K20(INT,INT),INT)) is non empty set
K20(NAT,NAT) is V5( RAT ) V5( INT ) non empty V35() V36() V37() V38() set
K20(K20(NAT,NAT),NAT) is V5( RAT ) V5( INT ) non empty V35() V36() V37() V38() set
K19(K20(K20(NAT,NAT),NAT)) is non empty set
1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
0 is 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
2 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
3 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
{} is functional empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V25() real ext-real non positive non negative integer V49() V50() V51() V52() V53() V54() V55() FinSequence-membered set
- 1 is V25() real ext-real non positive integer Element of REAL
sqrt 0 is V25() real ext-real Element of REAL
2 to_power 3 is V25() real ext-real Element of REAL
2 |^ 3 is set
8 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
4 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 to_power 4 is V25() real ext-real Element of REAL
2 |^ 4 is set
16 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
5 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 to_power 5 is V25() real ext-real Element of REAL
2 |^ 5 is set
32 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
6 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 to_power 6 is V25() real ext-real Element of REAL
2 |^ 6 is set
64 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
0 ! is V25() real ext-real Element of REAL
1 ! is V25() real ext-real Element of REAL
2 ! is V25() real ext-real Element of REAL
Funcs (NAT,REAL) is functional non empty FUNCTION_DOMAIN of NAT , REAL
c is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer set
2 to_power c is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 |^ c is set
c + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 to_power (c + 1) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 |^ (c + 1) is set
(c + 1) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 to_power 1 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 |^ 1 is set
(2 to_power c) * (2 to_power 1) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(2 to_power c) * 2 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(2 to_power c) + (2 to_power c) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(c + 1) + (2 to_power c) is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
0 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
x is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 to_power x is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 |^ x is set
x + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(x + 1) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(x + 1) + (2 to_power x) is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 to_power 2 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 |^ 2 is set
2 ^2 is V25() real ext-real Element of REAL
2 * 2 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer set
2 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
c is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer set
2 to_power c is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 |^ c is set
c + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
12 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
36 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
c is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
log (2,c) is V25() real ext-real Element of REAL
(log (2,c)) ^2 is V25() real ext-real Element of REAL
(log (2,c)) * (log (2,c)) is V25() real ext-real set
((log (2,c)) ^2) + 36 is V25() real ext-real Element of REAL
c is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() Element of K19(K20(NAT,REAL))
c . 0 is V25() real ext-real Element of REAL
x is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
c . x is V25() real ext-real Element of REAL
log (2,x) is V25() real ext-real Element of REAL
(log (2,x)) ^2 is V25() real ext-real Element of REAL
(log (2,x)) * (log (2,x)) is V25() real ext-real set
((log (2,x)) ^2) + 36 is V25() real ext-real Element of REAL
c is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() Element of K19(K20(NAT,REAL))
c . 0 is V25() real ext-real Element of REAL
c is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() Element of K19(K20(NAT,REAL))
c . 0 is V25() real ext-real Element of REAL
x is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
c . x is V25() real ext-real Element of REAL
log (2,x) is V25() real ext-real Element of REAL
(log (2,x)) ^2 is V25() real ext-real Element of REAL
(log (2,x)) * (log (2,x)) is V25() real ext-real set
((log (2,x)) ^2) + 36 is V25() real ext-real Element of REAL
0 + 0 is 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
f is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() Element of K19(K20(NAT,REAL))
f . 0 is V25() real ext-real Element of REAL
e is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() Element of K19(K20(NAT,REAL))
N0 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
e . N0 is V25() real ext-real Element of REAL
log (2,2) is V25() real ext-real Element of REAL
log (2,N0) is V25() real ext-real Element of REAL
N0 to_power 3 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
N0 |^ 3 is set
(N0 to_power 3) * (log (2,N0)) is V25() real ext-real Element of REAL
(N0 to_power 3) * 0 is 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
2 ^2 is V25() real ext-real Element of REAL
2 * 2 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer set
2 to_power 2 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 |^ 2 is set
log (2,4) is V25() real ext-real Element of REAL
log (2,2) is V25() real ext-real Element of REAL
2 * (log (2,2)) is V25() real ext-real Element of REAL
2 * 1 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
7 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
x is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() eventually-nonnegative eventually-positive eventually-nonzero Element of K19(K20(NAT,REAL))
N0 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer set
N0 ^2 is V25() real ext-real set
N0 * N0 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer set
7 * (N0 ^2) is V25() real ext-real Element of REAL
x . N0 is V25() real ext-real Element of REAL
N0 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(N0 + 1) ^2 is V25() real ext-real set
(N0 + 1) * (N0 + 1) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer set
7 * ((N0 + 1) ^2) is V25() real ext-real Element of REAL
x . (N0 + 1) is V25() real ext-real Element of REAL
log (2,(N0 + 1)) is V25() real ext-real Element of REAL
(log (2,(N0 + 1))) ^2 is V25() real ext-real Element of REAL
(log (2,(N0 + 1))) * (log (2,(N0 + 1))) is V25() real ext-real set
((log (2,(N0 + 1))) ^2) + 36 is V25() real ext-real Element of REAL
2 to_power N0 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 |^ N0 is set
N0 + 0 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
log (2,(2 to_power N0)) is V25() real ext-real Element of REAL
log (2,N0) is V25() real ext-real set
N0 * (log (2,2)) is V25() real ext-real Element of REAL
N0 * 1 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
14 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
14 * N0 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 * (log (2,N0)) is V25() real ext-real Element of REAL
7 * 2 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(7 * 2) * N0 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
((7 * 2) * N0) + 7 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(2 * (log (2,N0))) + 1 is V25() real ext-real Element of REAL
(log (2,N0)) ^2 is V25() real ext-real set
(log (2,N0)) * (log (2,N0)) is V25() real ext-real set
2 * N0 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
7 * (2 * N0) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(7 * (2 * N0)) + 7 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
((log (2,N0)) ^2) + ((7 * (2 * N0)) + 7) is V25() real ext-real Element of REAL
((log (2,N0)) ^2) + ((2 * (log (2,N0))) + 1) is V25() real ext-real Element of REAL
N0 + N0 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer set
log (2,(N0 + N0)) is V25() real ext-real set
log (2,(2 * N0)) is V25() real ext-real Element of REAL
(log (2,N0)) + (log (2,2)) is V25() real ext-real Element of REAL
(log (2,(N0 + N0))) ^2 is V25() real ext-real set
(log (2,(N0 + N0))) * (log (2,(N0 + N0))) is V25() real ext-real set
(log (2,N0)) + 1 is V25() real ext-real Element of REAL
((log (2,N0)) + 1) ^2 is V25() real ext-real Element of REAL
((log (2,N0)) + 1) * ((log (2,N0)) + 1) is V25() real ext-real set
((log (2,N0)) ^2) + (2 * (log (2,N0))) is V25() real ext-real Element of REAL
(((log (2,N0)) ^2) + (2 * (log (2,N0)))) + 1 is V25() real ext-real Element of REAL
(((log (2,N0)) ^2) + ((7 * (2 * N0)) + 7)) + 36 is V25() real ext-real Element of REAL
((log (2,(N0 + N0))) ^2) + 36 is V25() real ext-real Element of REAL
4 + 0 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(N0 + 1) ^2 is V25() real ext-real Element of REAL
7 * ((N0 + 1) ^2) is V25() real ext-real Element of REAL
7 * 1 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(7 * (2 * N0)) + (7 * 1) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(7 * (N0 ^2)) + ((7 * (2 * N0)) + (7 * 1)) is V25() real ext-real Element of REAL
(x . N0) + ((7 * (2 * N0)) + (7 * 1)) is V25() real ext-real Element of REAL
((log (2,N0)) ^2) + 36 is V25() real ext-real Element of REAL
x . 4 is V25() real ext-real Element of REAL
(2 ^2) + 36 is V25() real ext-real Element of REAL
40 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
4 ^2 is V25() real ext-real set
4 * 4 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer set
7 * (4 ^2) is V25() real ext-real Element of REAL
N0 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
N0 ^2 is V25() real ext-real Element of REAL
N0 * N0 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer set
7 * (N0 ^2) is V25() real ext-real Element of REAL
x . N0 is V25() real ext-real Element of REAL
log (2,3) is V25() real ext-real Element of REAL
t is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
f . t is V25() real ext-real Element of REAL
t to_power 2 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
t |^ 2 is set
t to_power 3 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
t |^ 3 is set
log (2,t) is V25() real ext-real Element of REAL
(t to_power 3) * (log (2,t)) is V25() real ext-real Element of REAL
(t to_power 2) * 1 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
12 * ((t to_power 3) * (log (2,t))) is V25() real ext-real Element of REAL
5 * (t to_power 2) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
12 * (t to_power 3) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(12 * (t to_power 3)) * (log (2,t)) is V25() real ext-real Element of REAL
t ^2 is V25() real ext-real Element of REAL
t * t is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer set
5 * (t ^2) is V25() real ext-real Element of REAL
(5 * (t ^2)) + 0 is V25() real ext-real Element of REAL
((12 * (t to_power 3)) * (log (2,t))) - (5 * (t ^2)) is V25() real ext-real Element of REAL
- (5 * (t ^2)) is V25() real ext-real set
((12 * (t to_power 3)) * (log (2,t))) + (- (5 * (t ^2))) is V25() real ext-real set
(log (2,t)) ^2 is V25() real ext-real Element of REAL
(log (2,t)) * (log (2,t)) is V25() real ext-real set
(((12 * (t to_power 3)) * (log (2,t))) - (5 * (t ^2))) + ((log (2,t)) ^2) is V25() real ext-real Element of REAL
0 + 0 is 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
((((12 * (t to_power 3)) * (log (2,t))) - (5 * (t ^2))) + ((log (2,t)) ^2)) + 36 is V25() real ext-real Element of REAL
t is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() eventually-nonnegative eventually-positive eventually-nonzero Element of K19(K20(NAT,REAL))
N0 is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() eventually-nonnegative eventually-positive eventually-nonzero Element of K19(K20(NAT,REAL))
Big_Oh N0 is functional non empty FUNCTION_DOMAIN of NAT , REAL
{ b₁ where b₁ is V1() V4( NAT ) V5( REAL ) Function-like V14( NAT ) quasi_total V35() V36() V37() Element of Funcs (NAT,REAL) : ex b₂ being V25() real ext-real Element of REAL ex b₃ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT st
( not b₂ <= 0 & ( for b₄ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT holds
( not b₃ <= b₄ or ( b₁ . b₄ <= b₂ * (N0 . b₄) & 0 <= b₁ . b₄ ) ) ) ) } is set
d is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
d to_power 3 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
d |^ 3 is set
12 * (d to_power 3) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
log (2,d) is V25() real ext-real Element of REAL
(12 * (d to_power 3)) * (log (2,d)) is V25() real ext-real Element of REAL
d ^2 is V25() real ext-real Element of REAL
d * d is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer set
5 * (d ^2) is V25() real ext-real Element of REAL
((12 * (d to_power 3)) * (log (2,d))) - (5 * (d ^2)) is V25() real ext-real Element of REAL
- (5 * (d ^2)) is V25() real ext-real set
((12 * (d to_power 3)) * (log (2,d))) + (- (5 * (d ^2))) is V25() real ext-real set
d is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() Element of K19(K20(NAT,REAL))
d . 0 is V25() real ext-real Element of REAL
N1 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
d . N1 is V25() real ext-real Element of REAL
N1 to_power 3 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
N1 |^ 3 is set
12 * (N1 to_power 3) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
log (2,N1) is V25() real ext-real Element of REAL
(12 * (N1 to_power 3)) * (log (2,N1)) is V25() real ext-real Element of REAL
N1 ^2 is V25() real ext-real Element of REAL
N1 * N1 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer set
5 * (N1 ^2) is V25() real ext-real Element of REAL
((12 * (N1 to_power 3)) * (log (2,N1))) - (5 * (N1 ^2)) is V25() real ext-real Element of REAL
- (5 * (N1 ^2)) is V25() real ext-real set
((12 * (N1 to_power 3)) * (log (2,N1))) + (- (5 * (N1 ^2))) is V25() real ext-real set
d is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() Element of K19(K20(NAT,REAL))
d . 0 is V25() real ext-real Element of REAL
d is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() Element of K19(K20(NAT,REAL))
d . 0 is V25() real ext-real Element of REAL
log (2,3) is V25() real ext-real Element of REAL
N1 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
d . N1 is V25() real ext-real Element of REAL
N1 to_power 2 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
N1 |^ 2 is set
N1 to_power 3 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
N1 |^ 3 is set
log (2,N1) is V25() real ext-real Element of REAL
(N1 to_power 3) * (log (2,N1)) is V25() real ext-real Element of REAL
(N1 to_power 2) * 1 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
12 * ((N1 to_power 3) * (log (2,N1))) is V25() real ext-real Element of REAL
5 * (N1 to_power 2) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
12 * (N1 to_power 3) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(12 * (N1 to_power 3)) * (log (2,N1)) is V25() real ext-real Element of REAL
N1 ^2 is V25() real ext-real Element of REAL
N1 * N1 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer set
5 * (N1 ^2) is V25() real ext-real Element of REAL
(5 * (N1 ^2)) + 0 is V25() real ext-real Element of REAL
((12 * (N1 to_power 3)) * (log (2,N1))) - (5 * (N1 ^2)) is V25() real ext-real Element of REAL
- (5 * (N1 ^2)) is V25() real ext-real set
((12 * (N1 to_power 3)) * (log (2,N1))) + (- (5 * (N1 ^2))) is V25() real ext-real set
N1 is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() eventually-nonnegative eventually-positive eventually-nonzero Element of K19(K20(NAT,REAL))
max (N1,x) is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() eventually-nonnegative eventually-positive eventually-nonzero Element of K19(K20(NAT,REAL))
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
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
N1 . n is V25() real ext-real Element of REAL
n ^2 is V25() real ext-real Element of REAL
n * n is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer set
7 * (n ^2) is V25() real ext-real Element of REAL
n to_power 3 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
n |^ 3 is set
n to_power 2 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
n |^ 2 is set
log (2,n) is V25() real ext-real Element of REAL
(n to_power 3) * (log (2,n)) is V25() real ext-real Element of REAL
(n to_power 2) * 1 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
12 * ((n to_power 3) * (log (2,n))) is V25() real ext-real Element of REAL
12 * (n to_power 2) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
12 * (n to_power 3) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(12 * (n to_power 3)) * (log (2,n)) is V25() real ext-real Element of REAL
12 * (n ^2) is V25() real ext-real Element of REAL
5 * (n ^2) is V25() real ext-real Element of REAL
((12 * (n to_power 3)) * (log (2,n))) - (5 * (n ^2)) is V25() real ext-real Element of REAL
- (5 * (n ^2)) is V25() real ext-real set
((12 * (n to_power 3)) * (log (2,n))) + (- (5 * (n ^2))) is V25() real ext-real set
(12 * (n ^2)) - (5 * (n ^2)) is V25() real ext-real Element of REAL
(12 * (n ^2)) + (- (5 * (n ^2))) is V25() real ext-real set
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
N1 . n is V25() real ext-real Element of REAL
x . n is V25() real ext-real Element of REAL
n ^2 is V25() real ext-real Element of REAL
n * n is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer set
7 * (n ^2) is V25() real ext-real Element of REAL
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
(max (N1,x)) . n is V25() real ext-real Element of REAL
N1 . n is V25() real ext-real Element of REAL
x . n is V25() real ext-real Element of REAL
max ((N1 . n),(x . n)) is V25() real ext-real Element of REAL
max (4,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
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
(max (N1,x)) . n is V25() real ext-real Element of REAL
N1 . n is V25() real ext-real Element of REAL
n to_power 3 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
n |^ 3 is set
12 * (n to_power 3) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
log (2,n) is V25() real ext-real Element of REAL
(12 * (n to_power 3)) * (log (2,n)) is V25() real ext-real Element of REAL
n ^2 is V25() real ext-real Element of REAL
n * n is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer set
5 * (n ^2) is V25() real ext-real Element of REAL
((12 * (n to_power 3)) * (log (2,n))) - (5 * (n ^2)) is V25() real ext-real Element of REAL
- (5 * (n ^2)) is V25() real ext-real set
((12 * (n to_power 3)) * (log (2,n))) + (- (5 * (n ^2))) is V25() real ext-real set
((12 * (n to_power 3)) * (log (2,n))) - 0 is V25() real ext-real Element of REAL
- 0 is functional empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V25() real ext-real non positive non negative integer V49() V50() V51() V52() V53() V54() V55() FinSequence-membered set
((12 * (n to_power 3)) * (log (2,n))) + (- 0) is V25() real ext-real set
(n to_power 3) * (log (2,n)) is V25() real ext-real Element of REAL
12 * ((n to_power 3) * (log (2,n))) is V25() real ext-real Element of REAL
N0 . n is V25() real ext-real Element of REAL
12 * (N0 . n) is V25() real ext-real Element of REAL
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
t . n is V25() real ext-real Element of REAL
N1 . n is V25() real ext-real Element of REAL
x . n is V25() real ext-real Element of REAL
(N1 . n) + (x . n) is V25() real ext-real Element of REAL
n to_power 3 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
n |^ 3 is set
12 * (n to_power 3) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
log (2,n) is V25() real ext-real Element of REAL
(12 * (n to_power 3)) * (log (2,n)) is V25() real ext-real Element of REAL
n ^2 is V25() real ext-real Element of REAL
n * n is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer set
5 * (n ^2) is V25() real ext-real Element of REAL
((12 * (n to_power 3)) * (log (2,n))) - (5 * (n ^2)) is V25() real ext-real Element of REAL
- (5 * (n ^2)) is V25() real ext-real set
((12 * (n to_power 3)) * (log (2,n))) + (- (5 * (n ^2))) is V25() real ext-real set
(log (2,n)) ^2 is V25() real ext-real Element of REAL
(log (2,n)) * (log (2,n)) is V25() real ext-real set
((log (2,n)) ^2) + 36 is V25() real ext-real Element of REAL
(((12 * (n to_power 3)) * (log (2,n))) - (5 * (n ^2))) + (((log (2,n)) ^2) + 36) is V25() real ext-real Element of REAL
(((12 * (n to_power 3)) * (log (2,n))) - (5 * (n ^2))) + ((log (2,n)) ^2) is V25() real ext-real Element of REAL
((((12 * (n to_power 3)) * (log (2,n))) - (5 * (n ^2))) + ((log (2,n)) ^2)) + 36 is V25() real ext-real Element of REAL
Big_Oh t is functional non empty FUNCTION_DOMAIN of NAT , REAL
{ b₁ where b₁ is V1() V4( NAT ) V5( REAL ) Function-like V14( NAT ) quasi_total V35() V36() V37() Element of Funcs (NAT,REAL) : ex b₂ being V25() real ext-real Element of REAL ex b₃ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT st
( not b₂ <= 0 & ( for b₄ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT holds
( not b₃ <= b₄ or ( b₁ . b₄ <= b₂ * (t . b₄) & 0 <= b₁ . b₄ ) ) ) ) } is set
N1 + x is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() eventually-nonnegative Element of K19(K20(NAT,REAL))
Big_Oh (N1 + x) is functional non empty FUNCTION_DOMAIN of NAT , REAL
{ b₁ where b₁ is V1() V4( NAT ) V5( REAL ) Function-like V14( NAT ) quasi_total V35() V36() V37() Element of Funcs (NAT,REAL) : ex b₂ being V25() real ext-real Element of REAL ex b₃ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT st
( not b₂ <= 0 & ( for b₄ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT holds
( not b₃ <= b₄ or ( b₁ . b₄ <= b₂ * ((N1 + x) . b₄) & 0 <= b₁ . b₄ ) ) ) ) } is set
Big_Oh (max (N1,x)) is functional non empty FUNCTION_DOMAIN of NAT , REAL
{ b₁ where b₁ is V1() V4( NAT ) V5( REAL ) Function-like V14( NAT ) quasi_total V35() V36() V37() Element of Funcs (NAT,REAL) : ex b₂ being V25() real ext-real Element of REAL ex b₃ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT st
( not b₂ <= 0 & ( for b₄ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT holds
( not b₃ <= b₄ or ( b₁ . b₄ <= b₂ * ((max (N1,x)) . b₄) & 0 <= b₁ . b₄ ) ) ) ) } is set
c is V25() real ext-real logbase Element of REAL
x is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() Element of K19(K20(NAT,REAL))
[/c\] is V25() real ext-real integer set
log (c,c) is V25() real ext-real Element of REAL
e is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
log (c,e) is V25() real ext-real Element of REAL
e + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
N0 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
e + 0 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
log (c,N0) is V25() real ext-real Element of REAL
x . N0 is V25() real ext-real Element of REAL
c is V25() real ext-real logbase Element of REAL
x is V25() real ext-real logbase Element of REAL
f is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() Element of K19(K20(NAT,REAL))
e is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() Element of K19(K20(NAT,REAL))
t is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() eventually-nonnegative eventually-positive eventually-nonzero Element of K19(K20(NAT,REAL))
Big_Oh t is functional non empty FUNCTION_DOMAIN of NAT , REAL
{ b₁ where b₁ is V1() V4( NAT ) V5( REAL ) Function-like V14( NAT ) quasi_total V35() V36() V37() Element of Funcs (NAT,REAL) : ex b₂ being V25() real ext-real Element of REAL ex b₃ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT st
( not b₂ <= 0 & ( for b₄ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT holds
( not b₃ <= b₄ or ( b₁ . b₄ <= b₂ * (t . b₄) & 0 <= b₁ . b₄ ) ) ) ) } is set
N0 is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() eventually-nonnegative eventually-positive eventually-nonzero Element of K19(K20(NAT,REAL))
Big_Oh N0 is functional non empty FUNCTION_DOMAIN of NAT , REAL
{ b₁ where b₁ is V1() V4( NAT ) V5( REAL ) Function-like V14( NAT ) quasi_total V35() V36() V37() Element of Funcs (NAT,REAL) : ex b₂ being V25() real ext-real Element of REAL ex b₃ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT st
( not b₂ <= 0 & ( for b₄ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT holds
( not b₃ <= b₄ or ( b₁ . b₄ <= b₂ * (N0 . b₄) & 0 <= b₁ . b₄ ) ) ) ) } is set
d is set
N1 is V1() V4( NAT ) V5( REAL ) Function-like V14( NAT ) quasi_total V35() V36() V37() Element of Funcs (NAT,REAL)
b is V25() real ext-real Element of REAL
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
b is V25() real ext-real Element of REAL
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
N + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
n is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
g9 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
t . g9 is V25() real ext-real Element of REAL
log (c,g9) is V25() real ext-real Element of REAL
log (c,x) is V25() real ext-real Element of REAL
log (x,g9) is V25() real ext-real Element of REAL
(log (c,x)) * (log (x,g9)) is V25() real ext-real Element of REAL
N + 0 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
N1 . g9 is V25() real ext-real Element of REAL
b * (t . g9) is V25() real ext-real Element of REAL
b * (log (c,x)) is V25() real ext-real Element of REAL
(b * (log (c,x))) * (log (x,g9)) is V25() real ext-real Element of REAL
N0 . g9 is V25() real ext-real Element of REAL
(b * (log (c,x))) * (N0 . g9) is V25() real ext-real Element of REAL
log (c,1) is V25() real ext-real Element of REAL
b * 0 is V25() real ext-real Element of REAL
n is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
N1 is V1() V4( NAT ) V5( REAL ) Function-like V14( NAT ) quasi_total V35() V36() V37() Element of Funcs (NAT,REAL)
b is V25() real ext-real Element of REAL
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
b is V25() real ext-real Element of REAL
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
N + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
n is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
g9 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
N0 . g9 is V25() real ext-real Element of REAL
log (x,g9) is V25() real ext-real Element of REAL
log (x,c) is V25() real ext-real Element of REAL
log (c,g9) is V25() real ext-real Element of REAL
(log (x,c)) * (log (c,g9)) is V25() real ext-real Element of REAL
N + 0 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
N1 . g9 is V25() real ext-real Element of REAL
b * (N0 . g9) is V25() real ext-real Element of REAL
b * (log (x,c)) is V25() real ext-real Element of REAL
(b * (log (x,c))) * (log (c,g9)) is V25() real ext-real Element of REAL
t . g9 is V25() real ext-real Element of REAL
(b * (log (x,c))) * (t . g9) is V25() real ext-real Element of REAL
log (x,1) is V25() real ext-real Element of REAL
b * 0 is V25() real ext-real Element of REAL
n is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
c is V25() real ext-real Element of REAL
x is V25() real ext-real Element of REAL
f is V25() real ext-real Element of REAL
e is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() Element of K19(K20(NAT,REAL))
N0 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
e . N0 is V25() real ext-real Element of REAL
x * N0 is V25() real ext-real Element of REAL
(x * N0) + f is V25() real ext-real Element of REAL
c to_power ((x * N0) + f) is V25() real ext-real Element of REAL
e is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() Element of K19(K20(NAT,REAL))
N0 is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() Element of K19(K20(NAT,REAL))
t is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
e . t is V25() real ext-real Element of REAL
x * t is V25() real ext-real Element of REAL
(x * t) + f is V25() real ext-real Element of REAL
c to_power ((x * t) + f) is V25() real ext-real Element of REAL
N0 . t is V25() real ext-real Element of REAL
c is non empty V25() real ext-real positive non negative Element of REAL
x is V25() real ext-real Element of REAL
f is V25() real ext-real Element of REAL
(c,x,f) is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() Element of K19(K20(NAT,REAL))
N0 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(c,x,f) . N0 is V25() real ext-real Element of REAL
x * N0 is V25() real ext-real Element of REAL
(x * N0) + f is V25() real ext-real Element of REAL
c to_power ((x * N0) + f) is V25() real ext-real Element of REAL
c is V25() real ext-real Element of REAL
f is V25() real ext-real Element of REAL
x is V25() real ext-real Element of REAL
c to_power x is V25() real ext-real Element of REAL
log (f,c) is V25() real ext-real Element of REAL
x * (log (f,c)) is V25() real ext-real Element of REAL
f to_power (x * (log (f,c))) is V25() real ext-real Element of REAL
log (f,(c to_power x)) is V25() real ext-real Element of REAL
x is non empty V25() real ext-real positive non negative Element of REAL
c is non empty V25() real ext-real positive non negative Element of REAL
(x,1,0) is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() eventually-nonnegative eventually-positive eventually-nonzero Element of K19(K20(NAT,REAL))
(c,1,0) is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() eventually-nonnegative eventually-positive eventually-nonzero Element of K19(K20(NAT,REAL))
Big_Oh (c,1,0) is functional non empty FUNCTION_DOMAIN of NAT , REAL
{ b₁ where b₁ is V1() V4( NAT ) V5( REAL ) Function-like V14( NAT ) quasi_total V35() V36() V37() Element of Funcs (NAT,REAL) : ex b₂ being V25() real ext-real Element of REAL ex b₃ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT st
( not b₂ <= 0 & ( for b₄ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT holds
( not b₃ <= b₄ or ( b₁ . b₄ <= b₂ * ((c,1,0) . b₄) & 0 <= b₁ . b₄ ) ) ) ) } is set
log (2,x) is V25() real ext-real Element of REAL
log (2,c) is V25() real ext-real Element of REAL
(log (2,x)) - (log (2,c)) is V25() real ext-real Element of REAL
- (log (2,c)) is V25() real ext-real set
(log (2,x)) + (- (log (2,c))) is V25() real ext-real set
t is V1() V4( NAT ) V5( REAL ) Function-like V14( NAT ) quasi_total V35() V36() V37() Element of Funcs (NAT,REAL)
d is V25() real ext-real Element of REAL
N1 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
d is V25() real ext-real Element of REAL
N1 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
log (2,d) is V25() real ext-real Element of REAL
(log (2,d)) / ((log (2,x)) - (log (2,c))) is V25() real ext-real Element of REAL
((log (2,x)) - (log (2,c))) " is V25() real ext-real set
(log (2,d)) * (((log (2,x)) - (log (2,c))) ") is V25() real ext-real set
[/((log (2,d)) / ((log (2,x)) - (log (2,c))))\] is V25() real ext-real integer set
max (N1,[/((log (2,d)) / ((log (2,x)) - (log (2,c))))\]) is V25() real ext-real set
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
n + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(n + 1) * (log (2,c)) is V25() real ext-real Element of REAL
2 to_power ((n + 1) * (log (2,c))) is V25() real ext-real Element of REAL
(log (2,c)) + 0 is V25() real ext-real Element of REAL
n + 0 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(n + 1) * ((log (2,x)) - (log (2,c))) is V25() real ext-real Element of REAL
((log (2,d)) / ((log (2,x)) - (log (2,c)))) * ((log (2,x)) - (log (2,c))) is V25() real ext-real Element of REAL
2 to_power ((n + 1) * ((log (2,x)) - (log (2,c)))) is V25() real ext-real Element of REAL
2 to_power (log (2,d)) is V25() real ext-real Element of REAL
(n + 1) * (log (2,x)) is V25() real ext-real Element of REAL
((n + 1) * (log (2,x))) - ((n + 1) * (log (2,c))) is V25() real ext-real Element of REAL
- ((n + 1) * (log (2,c))) is V25() real ext-real set
((n + 1) * (log (2,x))) + (- ((n + 1) * (log (2,c)))) is V25() real ext-real set
2 to_power (((n + 1) * (log (2,x))) - ((n + 1) * (log (2,c)))) is V25() real ext-real Element of REAL
2 to_power ((n + 1) * (log (2,x))) is V25() real ext-real Element of REAL
(2 to_power ((n + 1) * (log (2,x)))) / (2 to_power ((n + 1) * (log (2,c)))) is V25() real ext-real Element of REAL
(2 to_power ((n + 1) * (log (2,c)))) " is V25() real ext-real set
(2 to_power ((n + 1) * (log (2,x)))) * ((2 to_power ((n + 1) * (log (2,c)))) ") is V25() real ext-real set
((2 to_power ((n + 1) * (log (2,x)))) / (2 to_power ((n + 1) * (log (2,c))))) * (2 to_power ((n + 1) * (log (2,c)))) is V25() real ext-real Element of REAL
d * (2 to_power ((n + 1) * (log (2,c)))) is V25() real ext-real Element of REAL
x to_power (n + 1) is V25() real ext-real Element of REAL
x |^ (n + 1) is set
c to_power (n + 1) is V25() real ext-real Element of REAL
c |^ (n + 1) is set
d * (c to_power (n + 1)) is V25() real ext-real Element of REAL
(x,1,0) . (n + 1) is V25() real ext-real Element of REAL
(c,1,0) . (n + 1) is V25() real ext-real Element of REAL
d * ((c,1,0) . (n + 1)) is V25() real ext-real Element of REAL
1 * (n + 1) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(1 * (n + 1)) + 0 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
x to_power ((1 * (n + 1)) + 0) is V25() real ext-real Element of REAL
x |^ ((1 * (n + 1)) + 0) is set
c is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
log (2,c) is V25() real ext-real Element of REAL
c is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() Element of K19(K20(NAT,REAL))
c . 0 is V25() real ext-real Element of REAL
x is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
c . x is V25() real ext-real Element of REAL
log (2,x) is V25() real ext-real Element of REAL
c is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() Element of K19(K20(NAT,REAL))
c . 0 is V25() real ext-real Element of REAL
x is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() Element of K19(K20(NAT,REAL))
x . 0 is V25() real ext-real Element of REAL
f is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
c . f is V25() real ext-real Element of REAL
x . f is V25() real ext-real Element of REAL
f is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
c . f is V25() real ext-real Element of REAL
log (2,f) is V25() real ext-real Element of REAL
x . f is V25() real ext-real Element of REAL
f is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
() is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() Element of K19(K20(NAT,REAL))
c is V25() real ext-real Element of REAL
x is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
x to_power c is V25() real ext-real Element of REAL
x is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() Element of K19(K20(NAT,REAL))
x . 0 is V25() real ext-real Element of REAL
f is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
x . f is V25() real ext-real Element of REAL
f to_power c is V25() real ext-real Element of REAL
x is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() Element of K19(K20(NAT,REAL))
x . 0 is V25() real ext-real Element of REAL
f is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() Element of K19(K20(NAT,REAL))
f . 0 is V25() real ext-real Element of REAL
e is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
x . e is V25() real ext-real Element of REAL
f . e is V25() real ext-real Element of REAL
e is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
x . e is V25() real ext-real Element of REAL
e to_power c is V25() real ext-real Element of REAL
f . e is V25() real ext-real Element of REAL
e is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
x is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
() . x is V25() real ext-real Element of REAL
log (2,2) is V25() real ext-real Element of REAL
log (2,x) is V25() real ext-real Element of REAL
c is V25() real ext-real Element of REAL
(c) is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() Element of K19(K20(NAT,REAL))
f is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(c) . f is V25() real ext-real Element of REAL
f to_power c is V25() real ext-real Element of REAL
c is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() Element of K19(K20(NAT,REAL))
x is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() Element of K19(K20(NAT,REAL))
c /" x is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() Element of K19(K20(NAT,REAL))
x " is V1() V4( NAT ) Function-like V14( NAT ) V35() V36() V37() set
c (#) (x ") is V1() V4( NAT ) Function-like V14( NAT ) V35() V36() V37() set
f is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(c /" x) . f is V25() real ext-real Element of REAL
c . f is V25() real ext-real Element of REAL
x . f is V25() real ext-real Element of REAL
(c . f) / (x . f) is V25() real ext-real Element of REAL
(x . f) " is V25() real ext-real set
(c . f) * ((x . f) ") is V25() real ext-real set
x " is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() Element of K19(K20(NAT,REAL))
(x ") . f is V25() real ext-real Element of REAL
(c . f) * ((x ") . f) is V25() real ext-real Element of REAL
(x . f) " is V25() real ext-real Element of REAL
(c . f) * ((x . f) ") is V25() real ext-real Element of REAL
c is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() eventually-nonnegative Element of K19(K20(NAT,REAL))
x is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() eventually-nonnegative Element of K19(K20(NAT,REAL))
Big_Oh x is functional non empty FUNCTION_DOMAIN of NAT , REAL
{ b₁ where b₁ is V1() V4( NAT ) V5( REAL ) Function-like V14( NAT ) quasi_total V35() V36() V37() Element of Funcs (NAT,REAL) : ex b₂ being V25() real ext-real Element of REAL ex b₃ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT st
( not b₂ <= 0 & ( for b₄ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT holds
( not b₃ <= b₄ or ( b₁ . b₄ <= b₂ * (x . b₄) & 0 <= b₁ . b₄ ) ) ) ) } is set
Big_Oh c is functional non empty FUNCTION_DOMAIN of NAT , REAL
{ b₁ where b₁ is V1() V4( NAT ) V5( REAL ) Function-like V14( NAT ) quasi_total V35() V36() V37() Element of Funcs (NAT,REAL) : ex b₂ being V25() real ext-real Element of REAL ex b₃ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT st
( not b₂ <= 0 & ( for b₄ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT holds
( not b₃ <= b₄ or ( b₁ . b₄ <= b₂ * (c . b₄) & 0 <= b₁ . b₄ ) ) ) ) } is set
c is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() eventually-nonnegative Element of K19(K20(NAT,REAL))
Big_Oh c is functional non empty FUNCTION_DOMAIN of NAT , REAL
{ b₁ where b₁ is V1() V4( NAT ) V5( REAL ) Function-like V14( NAT ) quasi_total V35() V36() V37() Element of Funcs (NAT,REAL) : ex b₂ being V25() real ext-real Element of REAL ex b₃ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT st
( not b₂ <= 0 & ( for b₄ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT holds
( not b₃ <= b₄ or ( b₁ . b₄ <= b₂ * (c . b₄) & 0 <= b₁ . b₄ ) ) ) ) } is set
x is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() eventually-nonnegative Element of K19(K20(NAT,REAL))
Big_Oh x is functional non empty FUNCTION_DOMAIN of NAT , REAL
{ b₁ where b₁ is V1() V4( NAT ) V5( REAL ) Function-like V14( NAT ) quasi_total V35() V36() V37() Element of Funcs (NAT,REAL) : ex b₂ being V25() real ext-real Element of REAL ex b₃ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT st
( not b₂ <= 0 & ( for b₄ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT holds
( not b₃ <= b₄ or ( b₁ . b₄ <= b₂ * (x . b₄) & 0 <= b₁ . b₄ ) ) ) ) } is set
Big_Omega x is functional non empty FUNCTION_DOMAIN of NAT , REAL
{ b₁ where b₁ is V1() V4( NAT ) V5( REAL ) Function-like V14( NAT ) quasi_total V35() V36() V37() Element of Funcs (NAT,REAL) : ex b₂ being V25() real ext-real Element of REAL ex b₃ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT st
( not b₂ <= 0 & ( for b₄ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT holds
( not b₃ <= b₄ or ( b₂ * (x . b₄) <= b₁ . b₄ & 0 <= b₁ . b₄ ) ) ) ) } is set
f is set
e is V1() V4( NAT ) V5( REAL ) Function-like V14( NAT ) quasi_total V35() V36() V37() Element of Funcs (NAT,REAL)
N0 is V25() real ext-real Element of REAL
t is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
N0 is V25() real ext-real Element of REAL
t is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
d is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
N1 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
e . N1 is V25() real ext-real Element of REAL
d is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
d is V25() real ext-real Element of REAL
N1 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
c is V25() real ext-real set
x is V25() real ext-real set
f is V25() real ext-real set
c to_power f is V25() real ext-real set
x to_power f is V25() real ext-real set
c is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer set
2 to_power c is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 |^ c is set
2 * c is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(2 * c) + 3 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
c + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 to_power (c + 1) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 |^ (c + 1) is set
2 * (c + 1) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(2 * (c + 1)) + 3 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 to_power 1 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 |^ 1 is set
(2 to_power c) + (2 to_power c) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 + (2 to_power c) is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 + ((2 * c) + 3) is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 * (2 to_power c) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(2 to_power 1) * (2 to_power c) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 to_power 4 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 * 4 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(2 * 4) + 3 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
c is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer set
2 to_power c is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 |^ c is set
2 * c is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(2 * c) + 3 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
c is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer set
2 to_power c is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 |^ c is set
c + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(c + 1) ^2 is V25() real ext-real Element of REAL
(c + 1) * (c + 1) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer set
2 to_power (c + 1) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 |^ (c + 1) is set
(c + 1) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
((c + 1) + 1) ^2 is V25() real ext-real Element of REAL
((c + 1) + 1) * ((c + 1) + 1) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer set
2 * c is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(2 * c) + 3 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
((c + 1) ^2) + (2 to_power c) is V25() real ext-real Element of REAL
((c + 1) ^2) + ((2 * c) + 3) is V25() real ext-real Element of REAL
(2 to_power c) + (2 to_power c) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 * (2 to_power c) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(c + 1) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
((c + 1) + 1) ^2 is V25() real ext-real Element of REAL
((c + 1) + 1) * ((c + 1) + 1) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer set
2 to_power 1 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 |^ 1 is set
(2 to_power 1) * (2 to_power c) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 to_power 6 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
6 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(6 + 1) ^2 is V25() real ext-real Element of REAL
(6 + 1) * (6 + 1) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer set
c is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 to_power c is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 |^ c is set
c + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(c + 1) ^2 is V25() real ext-real Element of REAL
(c + 1) * (c + 1) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer set
6 - 1 is V25() real ext-real integer Element of REAL
- 1 is V25() real ext-real non positive integer set
6 + (- 1) is V25() real ext-real integer set
c is V25() real ext-real Element of REAL
2 to_power c is V25() real ext-real Element of REAL
c ^2 is V25() real ext-real Element of REAL
c * c is V25() real ext-real set
[\c/] is V25() real ext-real integer set
[/c\] is V25() real ext-real integer set
e is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
e + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
e + 0 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(e + 1) ^2 is V25() real ext-real Element of REAL
(e + 1) * (e + 1) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer set
e ^2 is V25() real ext-real Element of REAL
e * e is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer set
2 to_power e is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 |^ e is set
[\c/] + 1 is V25() real ext-real integer Element of REAL
[/c\] - 1 is V25() real ext-real integer Element of REAL
[/c\] + (- 1) is V25() real ext-real integer set
e is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
N0 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 to_power N0 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 |^ N0 is set
e ^2 is V25() real ext-real Element of REAL
e * e is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer set
5 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
c is non empty V25() real ext-real positive non negative Element of REAL
(c) is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() eventually-nonnegative eventually-positive eventually-nonzero Element of K19(K20(NAT,REAL))
x is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() Element of K19(K20(NAT,REAL))
x /" (c) is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() Element of K19(K20(NAT,REAL))
(c) " is V1() V4( NAT ) Function-like V14( NAT ) V35() V36() V37() set
x (#) ((c) ") is V1() V4( NAT ) Function-like V14( NAT ) V35() V36() V37() set
lim (x /" (c)) is V25() real ext-real Element of REAL
N0 is V25() real ext-real set
7 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
t is V25() real ext-real Element of REAL
7 / t is V25() real ext-real Element of REAL
t " is V25() real ext-real set
7 * (t ") is V25() real ext-real set
1 / c is V25() real ext-real non negative Element of REAL
c " is non empty V25() real ext-real positive non negative set
1 * (c ") is V25() real ext-real non negative set
(7 / t) to_power (1 / c) is V25() real ext-real Element of REAL
[/((7 / t) to_power (1 / c))\] is V25() real ext-real integer set
2 / c is V25() real ext-real non negative Element of REAL
2 * (c ") is V25() real ext-real non negative set
- (2 / c) is V25() real ext-real non positive Element of REAL
t to_power (- (2 / c)) is V25() real ext-real Element of REAL
(t to_power (- (2 / c))) + 1 is V25() real ext-real Element of REAL
[/((t to_power (- (2 / c))) + 1)\] is V25() real ext-real integer set
max ([/((7 / t) to_power (1 / c))\],[/((t to_power (- (2 / c))) + 1)\]) is V25() real ext-real set
max ((max ([/((7 / t) to_power (1 / c))\],[/((t to_power (- (2 / c))) + 1)\])),2) is V25() real ext-real set
N0 to_power (- (2 / c)) is V25() real ext-real set
(N0 to_power (- (2 / c))) + 1 is V25() real ext-real Element of REAL
(N0 to_power (- (2 / c))) + 0 is V25() real ext-real Element of REAL
t to_power 2 is V25() real ext-real Element of REAL
t |^ 2 is set
7 / N0 is V25() real ext-real Element of REAL
N0 " is V25() real ext-real set
7 * (N0 ") is V25() real ext-real set
(7 / N0) to_power (1 / c) is V25() real ext-real Element of REAL
7 * (N0 ") is V25() real ext-real Element of REAL
7 * 0 is 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
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
g9 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
g9 to_power c is V25() real ext-real Element of REAL
t * (g9 to_power c) is V25() real ext-real Element of REAL
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
((7 / N0) to_power (1 / c)) to_power c is V25() real ext-real Element of REAL
c * (1 / c) is V25() real ext-real non negative Element of REAL
(7 / t) to_power (c * (1 / c)) is V25() real ext-real Element of REAL
(7 / N0) to_power 1 is V25() real ext-real Element of REAL
(7 / N0) |^ 1 is set
(7 / N0) * N0 is V25() real ext-real Element of REAL
N0 * (g9 to_power c) is V25() real ext-real Element of REAL
2 to_power (t * (g9 to_power c)) is V25() real ext-real Element of REAL
(t * (g9 to_power c)) ^2 is V25() real ext-real Element of REAL
(t * (g9 to_power c)) * (t * (g9 to_power c)) is V25() real ext-real set
(N0 to_power (- (2 / c))) to_power c is V25() real ext-real set
(- (2 / c)) * c is V25() real ext-real non positive Element of REAL
t to_power ((- (2 / c)) * c) is V25() real ext-real Element of REAL
(2 / c) * c is V25() real ext-real non negative Element of REAL
- ((2 / c) * c) is V25() real ext-real non positive Element of REAL
N0 to_power (- ((2 / c) * c)) is V25() real ext-real set
- 2 is V25() real ext-real non positive integer Element of REAL
N0 to_power (- 2) is V25() real ext-real set
N0 to_power 2 is V25() real ext-real set
N0 |^ 2 is set
(N0 to_power 2) * (g9 to_power c) is V25() real ext-real Element of REAL
(N0 to_power 2) * (N0 to_power (- 2)) is V25() real ext-real set
(t to_power 2) * (g9 to_power c) is V25() real ext-real Element of REAL
2 + (- 2) is V25() real ext-real integer Element of REAL
t to_power (2 + (- 2)) is V25() real ext-real Element of REAL
t ^2 is V25() real ext-real Element of REAL
t * t is V25() real ext-real set
(t ^2) * (g9 to_power c) is V25() real ext-real Element of REAL
1 / 1 is V25() real ext-real non negative Element of REAL
1 " is non empty V25() real ext-real positive non negative set
1 * (1 ") is V25() real ext-real non negative set
N0 * (t * (g9 to_power c)) is V25() real ext-real Element of REAL
1 / (N0 * (t * (g9 to_power c))) is V25() real ext-real Element of REAL
(N0 * (t * (g9 to_power c))) " is V25() real ext-real set
1 * ((N0 * (t * (g9 to_power c))) ") is V25() real ext-real set
(2 to_power (t * (g9 to_power c))) * 1 is V25() real ext-real Element of REAL
(t * (g9 to_power c)) * N0 is V25() real ext-real Element of REAL
(2 to_power (t * (g9 to_power c))) / ((t * (g9 to_power c)) * N0) is V25() real ext-real Element of REAL
((t * (g9 to_power c)) * N0) " is V25() real ext-real set
(2 to_power (t * (g9 to_power c))) * (((t * (g9 to_power c)) * N0) ") is V25() real ext-real set
N0 * 0 is V25() real ext-real Element of REAL
(2 to_power (t * (g9 to_power c))) / (t * (g9 to_power c)) is V25() real ext-real Element of REAL
(t * (g9 to_power c)) " is V25() real ext-real set
(2 to_power (t * (g9 to_power c))) * ((t * (g9 to_power c)) ") is V25() real ext-real set
((2 to_power (t * (g9 to_power c))) / (t * (g9 to_power c))) / N0 is V25() real ext-real Element of REAL
((2 to_power (t * (g9 to_power c))) / (t * (g9 to_power c))) * (N0 ") is V25() real ext-real set
log (2,(2 to_power (t * (g9 to_power c)))) is V25() real ext-real Element of REAL
log (2,(g9 to_power c)) is V25() real ext-real Element of REAL
log (2,2) is V25() real ext-real Element of REAL
(t * (g9 to_power c)) * (log (2,2)) is V25() real ext-real Element of REAL
(t * (g9 to_power c)) * 1 is V25() real ext-real Element of REAL
(log (2,(g9 to_power c))) / (g9 to_power c) is V25() real ext-real Element of REAL
(g9 to_power c) " is V25() real ext-real set
(log (2,(g9 to_power c))) * ((g9 to_power c) ") is V25() real ext-real set
g9 to_power 0 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
g9 |^ 0 is set
log (2,1) is V25() real ext-real Element of REAL
(x /" (c)) . g9 is V25() real ext-real Element of REAL
x . g9 is V25() real ext-real Element of REAL
(c) . g9 is V25() real ext-real Element of REAL
(x . g9) / ((c) . g9) is V25() real ext-real Element of REAL
((c) . g9) " is V25() real ext-real set
(x . g9) * (((c) . g9) ") is V25() real ext-real set
(log (2,(g9 to_power c))) / ((c) . g9) is V25() real ext-real Element of REAL
(log (2,(g9 to_power c))) * (((c) . g9) ") is V25() real ext-real set
((x /" (c)) . g9) - 0 is V25() real ext-real Element of REAL
- 0 is functional empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V25() real ext-real non positive non negative integer V49() V50() V51() V52() V53() V54() V55() FinSequence-membered set
((x /" (c)) . g9) + (- 0) is V25() real ext-real set
abs (((x /" (c)) . g9) - 0) is V25() real ext-real Element of REAL
x is V25() real ext-real Element of REAL
(x) is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() eventually-nonnegative eventually-positive eventually-nonzero Element of K19(K20(NAT,REAL))
() /" (x) is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() Element of K19(K20(NAT,REAL))
(x) " is V1() V4( NAT ) Function-like V14( NAT ) V35() V36() V37() set
() (#) ((x) ") is V1() V4( NAT ) Function-like V14( NAT ) V35() V36() V37() set
lim (() /" (x)) is V25() real ext-real Element of REAL
f is non empty V25() real ext-real positive non negative Element of REAL
(f) is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() eventually-nonnegative eventually-positive eventually-nonzero Element of K19(K20(NAT,REAL))
() /" (f) is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() Element of K19(K20(NAT,REAL))
(f) " is V1() V4( NAT ) Function-like V14( NAT ) V35() V36() V37() set
() (#) ((f) ") is V1() V4( NAT ) Function-like V14( NAT ) V35() V36() V37() set
t is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
t to_power f is V25() real ext-real Element of REAL
log (2,(t to_power f)) is V25() real ext-real Element of REAL
t is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() Element of K19(K20(NAT,REAL))
t . 0 is V25() real ext-real Element of REAL
d is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
t . d is V25() real ext-real Element of REAL
d to_power f is V25() real ext-real Element of REAL
log (2,(d to_power f)) is V25() real ext-real Element of REAL
t is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() Element of K19(K20(NAT,REAL))
t . 0 is V25() real ext-real Element of REAL
t is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() Element of K19(K20(NAT,REAL))
t . 0 is V25() real ext-real Element of REAL
t /" (f) is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() Element of K19(K20(NAT,REAL))
t (#) ((f) ") is V1() V4( NAT ) Function-like V14( NAT ) V35() V36() V37() set
f / f is V25() real ext-real non negative Element of REAL
f " is non empty V25() real ext-real positive non negative set
f * (f ") is V25() real ext-real non negative set
1 / f is V25() real ext-real non negative Element of REAL
1 * (f ") is V25() real ext-real non negative set
f * (1 / f) is V25() real ext-real non negative Element of REAL
N1 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(() /" (f)) . N1 is V25() real ext-real Element of REAL
(t /" (f)) . N1 is V25() real ext-real Element of REAL
(1 / f) * ((t /" (f)) . N1) is V25() real ext-real Element of REAL
() . N1 is V25() real ext-real Element of REAL
(f) . N1 is V25() real ext-real Element of REAL
(() . N1) / ((f) . N1) is V25() real ext-real Element of REAL
((f) . N1) " is V25() real ext-real set
(() . N1) * (((f) . N1) ") is V25() real ext-real set
t . N1 is V25() real ext-real Element of REAL
(t . N1) / ((f) . N1) is V25() real ext-real Element of REAL
(t . N1) * (((f) . N1) ") is V25() real ext-real set
0 / ((f) . N1) is V25() real ext-real Element of REAL
0 * (((f) . N1) ") is V25() real ext-real set
0 * (1 / f) is functional empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V25() real ext-real non positive non negative integer V49() V50() V51() V52() V53() V54() V55() FinSequence-membered Element of REAL
N1 to_power f is V25() real ext-real Element of REAL
log (2,N1) is V25() real ext-real Element of REAL
(log (2,N1)) / ((f) . N1) is V25() real ext-real Element of REAL
(log (2,N1)) * (((f) . N1) ") is V25() real ext-real set
N1 to_power (f * (1 / f)) is V25() real ext-real Element of REAL
log (2,(N1 to_power (f * (1 / f)))) is V25() real ext-real Element of REAL
(log (2,(N1 to_power (f * (1 / f))))) / ((f) . N1) is V25() real ext-real Element of REAL
(log (2,(N1 to_power (f * (1 / f))))) * (((f) . N1) ") is V25() real ext-real set
(N1 to_power f) to_power (1 / f) is V25() real ext-real Element of REAL
log (2,((N1 to_power f) to_power (1 / f))) is V25() real ext-real Element of REAL
(log (2,((N1 to_power f) to_power (1 / f)))) / ((f) . N1) is V25() real ext-real Element of REAL
(log (2,((N1 to_power f) to_power (1 / f)))) * (((f) . N1) ") is V25() real ext-real set
log (2,(N1 to_power f)) is V25() real ext-real Element of REAL
(1 / f) * (log (2,(N1 to_power f))) is V25() real ext-real Element of REAL
((1 / f) * (log (2,(N1 to_power f)))) / ((f) . N1) is V25() real ext-real Element of REAL
((1 / f) * (log (2,(N1 to_power f)))) * (((f) . N1) ") is V25() real ext-real set
((f) . N1) " is V25() real ext-real Element of REAL
((1 / f) * (log (2,(N1 to_power f)))) * (((f) . N1) ") is V25() real ext-real Element of REAL
(log (2,(N1 to_power f))) * (((f) . N1) ") is V25() real ext-real Element of REAL
(1 / f) * ((log (2,(N1 to_power f))) * (((f) . N1) ")) is V25() real ext-real Element of REAL
(log (2,(N1 to_power f))) / ((f) . N1) is V25() real ext-real Element of REAL
(log (2,(N1 to_power f))) * (((f) . N1) ") is V25() real ext-real set
(1 / f) * ((log (2,(N1 to_power f))) / ((f) . N1)) is V25() real ext-real Element of REAL
(1 / f) (#) (t /" (f)) is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() Element of K19(K20(NAT,REAL))
lim (t /" (f)) is V25() real ext-real Element of REAL
lim (() /" (f)) is V25() real ext-real Element of REAL
lim ((1 / f) (#) (t /" (f))) is V25() real ext-real Element of REAL
(1 / f) * 0 is functional empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V25() real ext-real non positive non negative integer V49() V50() V51() V52() V53() V54() V55() FinSequence-membered Element of REAL
Big_Oh () is functional non empty FUNCTION_DOMAIN of NAT , REAL
{ b₁ where b₁ is V1() V4( NAT ) V5( REAL ) Function-like V14( NAT ) quasi_total V35() V36() V37() Element of Funcs (NAT,REAL) : ex b₂ being V25() real ext-real Element of REAL ex b₃ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT st
( not b₂ <= 0 & ( for b₄ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT holds
( not b₃ <= b₄ or ( b₁ . b₄ <= b₂ * (() . b₄) & 0 <= b₁ . b₄ ) ) ) ) } is set
1 / 2 is V25() real ext-real non negative Element of REAL
2 " is non empty V25() real ext-real positive non negative set
1 * (2 ") is V25() real ext-real non negative set
((1 / 2)) is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() eventually-nonnegative eventually-positive eventually-nonzero Element of K19(K20(NAT,REAL))
Big_Oh ((1 / 2)) is functional non empty FUNCTION_DOMAIN of NAT , REAL
{ b₁ where b₁ is V1() V4( NAT ) V5( REAL ) Function-like V14( NAT ) quasi_total V35() V36() V37() Element of Funcs (NAT,REAL) : ex b₂ being V25() real ext-real Element of REAL ex b₃ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT st
( not b₂ <= 0 & ( for b₄ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT holds
( not b₃ <= b₄ or ( b₁ . b₄ <= b₂ * (((1 / 2)) . b₄) & 0 <= b₁ . b₄ ) ) ) ) } is set
() /" ((1 / 2)) is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() Element of K19(K20(NAT,REAL))
((1 / 2)) " is V1() V4( NAT ) Function-like V14( NAT ) V35() V36() V37() set
() (#) (((1 / 2)) ") is V1() V4( NAT ) Function-like V14( NAT ) V35() V36() V37() set
lim (() /" ((1 / 2))) is V25() real ext-real Element of REAL
Big_Omega ((1 / 2)) is functional non empty FUNCTION_DOMAIN of NAT , REAL
{ b₁ where b₁ is V1() V4( NAT ) V5( REAL ) Function-like V14( NAT ) quasi_total V35() V36() V37() Element of Funcs (NAT,REAL) : ex b₂ being V25() real ext-real Element of REAL ex b₃ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT st
( not b₂ <= 0 & ( for b₄ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT holds
( not b₃ <= b₄ or ( b₂ * (((1 / 2)) . b₄) <= b₁ . b₄ & 0 <= b₁ . b₄ ) ) ) ) } is set
Big_Omega () is functional non empty FUNCTION_DOMAIN of NAT , REAL
{ b₁ where b₁ is V1() V4( NAT ) V5( REAL ) Function-like V14( NAT ) quasi_total V35() V36() V37() Element of Funcs (NAT,REAL) : ex b₂ being V25() real ext-real Element of REAL ex b₃ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT st
( not b₂ <= 0 & ( for b₄ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT holds
( not b₃ <= b₄ or ( b₂ * (() . b₄) <= b₁ . b₄ & 0 <= b₁ . b₄ ) ) ) ) } is set
Big_Omega ((1 / 2)) is functional non empty FUNCTION_DOMAIN of NAT , REAL
{ b₁ where b₁ is V1() V4( NAT ) V5( REAL ) Function-like V14( NAT ) quasi_total V35() V36() V37() Element of Funcs (NAT,REAL) : ex b₂ being V25() real ext-real Element of REAL ex b₃ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT st
( not b₂ <= 0 & ( for b₄ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT holds
( not b₃ <= b₄ or ( b₂ * (((1 / 2)) . b₄) <= b₁ . b₄ & 0 <= b₁ . b₄ ) ) ) ) } is set
c is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() Element of K19(K20(NAT,REAL))
x is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
Sum (c,x) is V25() real ext-real Element of REAL
x + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
Sum (c,(x + 1)) is V25() real ext-real Element of REAL
f is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
f + 0 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
c . f is V25() real ext-real Element of REAL
c . (x + 1) is V25() real ext-real Element of REAL
0 + 0 is 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
(Sum (c,x)) + (c . (x + 1)) is V25() real ext-real Element of REAL
f is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
c . f is V25() real ext-real Element of REAL
Partial_Sums c is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() Element of K19(K20(NAT,REAL))
(Partial_Sums c) . x is V25() real ext-real Element of REAL
((Partial_Sums c) . x) + (c . (x + 1)) is V25() real ext-real Element of REAL
(Partial_Sums c) . (x + 1) is V25() real ext-real Element of REAL
Sum (c,0) is V25() real ext-real Element of REAL
c . 0 is V25() real ext-real Element of REAL
Partial_Sums c is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() Element of K19(K20(NAT,REAL))
(Partial_Sums c) . 0 is V25() real ext-real Element of REAL
x is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
Sum (c,x) is V25() real ext-real Element of REAL
c is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() Element of K19(K20(NAT,REAL))
x is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() Element of K19(K20(NAT,REAL))
f is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
Sum (c,f) is V25() real ext-real Element of REAL
Sum (x,f) is V25() real ext-real Element of REAL
f + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
Sum (c,(f + 1)) is V25() real ext-real Element of REAL
Sum (x,(f + 1)) is V25() real ext-real Element of REAL
e is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
e + 0 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
c . e is V25() real ext-real Element of REAL
x . e is V25() real ext-real Element of REAL
c . (f + 1) is V25() real ext-real Element of REAL
x . (f + 1) is V25() real ext-real Element of REAL
(Sum (c,f)) + (c . (f + 1)) is V25() real ext-real Element of REAL
(Sum (x,f)) + (x . (f + 1)) is V25() real ext-real Element of REAL
e is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
c . e is V25() real ext-real Element of REAL
x . e is V25() real ext-real Element of REAL
Partial_Sums c is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() Element of K19(K20(NAT,REAL))
(Partial_Sums c) . f is V25() real ext-real Element of REAL
((Partial_Sums c) . f) + (c . (f + 1)) is V25() real ext-real Element of REAL
(Partial_Sums c) . (f + 1) is V25() real ext-real Element of REAL
Partial_Sums x is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() Element of K19(K20(NAT,REAL))
(Partial_Sums x) . f is V25() real ext-real Element of REAL
((Partial_Sums x) . f) + (x . (f + 1)) is V25() real ext-real Element of REAL
(Partial_Sums x) . (f + 1) is V25() real ext-real Element of REAL
Sum (c,0) is V25() real ext-real Element of REAL
Sum (x,0) is V25() real ext-real Element of REAL
c . 0 is V25() real ext-real Element of REAL
x . 0 is V25() real ext-real Element of REAL
Partial_Sums c is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() Element of K19(K20(NAT,REAL))
(Partial_Sums c) . 0 is V25() real ext-real Element of REAL
Partial_Sums x is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() Element of K19(K20(NAT,REAL))
(Partial_Sums x) . 0 is V25() real ext-real Element of REAL
f is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
Sum (c,f) is V25() real ext-real Element of REAL
Sum (x,f) is V25() real ext-real Element of REAL
c is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() Element of K19(K20(NAT,REAL))
c . 0 is V25() real ext-real Element of REAL
x is V25() real ext-real Element of REAL
f is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
Sum (c,f) is V25() real ext-real Element of REAL
x * f is V25() real ext-real Element of REAL
f + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
Sum (c,(f + 1)) is V25() real ext-real Element of REAL
x * (f + 1) is V25() real ext-real Element of REAL
Partial_Sums c is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() Element of K19(K20(NAT,REAL))
(Partial_Sums c) . (f + 1) is V25() real ext-real Element of REAL
(Partial_Sums c) . f is V25() real ext-real Element of REAL
c . (f + 1) is V25() real ext-real Element of REAL
((Partial_Sums c) . f) + (c . (f + 1)) is V25() real ext-real Element of REAL
(x * f) + (c . (f + 1)) is V25() real ext-real Element of REAL
x * 1 is V25() real ext-real Element of REAL
(x * f) + (x * 1) is V25() real ext-real Element of REAL
Partial_Sums c is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() Element of K19(K20(NAT,REAL))
(Partial_Sums c) . 0 is V25() real ext-real Element of REAL
Sum (c,0) is V25() real ext-real Element of REAL
x * 0 is V25() real ext-real Element of REAL
f is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
Sum (c,f) is V25() real ext-real Element of REAL
x * f is V25() real ext-real Element of REAL
c is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() Element of K19(K20(NAT,REAL))
x is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer set
f is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer set
Sum (c,x,f) is V25() real ext-real Element of REAL
x + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
c . (x + 1) is V25() real ext-real Element of REAL
(Sum (c,x,f)) + (c . (x + 1)) is V25() real ext-real Element of REAL
Sum (c,(x + 1),f) is V25() real ext-real Element of REAL
Sum (c,x) is V25() real ext-real Element of REAL
Sum (c,f) is V25() real ext-real Element of REAL
(Sum (c,x)) - (Sum (c,f)) is V25() real ext-real Element of REAL
- (Sum (c,f)) is V25() real ext-real set
(Sum (c,x)) + (- (Sum (c,f))) is V25() real ext-real set
((Sum (c,x)) - (Sum (c,f))) + (c . (x + 1)) is V25() real ext-real Element of REAL
(Sum (c,x)) + (c . (x + 1)) is V25() real ext-real Element of REAL
- (Sum (c,f)) is V25() real ext-real Element of REAL
((Sum (c,x)) + (c . (x + 1))) + (- (Sum (c,f))) is V25() real ext-real Element of REAL
Partial_Sums c is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() Element of K19(K20(NAT,REAL))
(Partial_Sums c) . x is V25() real ext-real Element of REAL
((Partial_Sums c) . x) + (c . (x + 1)) is V25() real ext-real Element of REAL
(((Partial_Sums c) . x) + (c . (x + 1))) + (- (Sum (c,f))) is V25() real ext-real Element of REAL
(Partial_Sums c) . (x + 1) is V25() real ext-real Element of REAL
((Partial_Sums c) . (x + 1)) + (- (Sum (c,f))) is V25() real ext-real Element of REAL
Sum (c,(x + 1)) is V25() real ext-real Element of REAL
(Sum (c,(x + 1))) + (- (Sum (c,f))) is V25() real ext-real Element of REAL
(Sum (c,(x + 1))) - (Sum (c,f)) is V25() real ext-real Element of REAL
(Sum (c,(x + 1))) + (- (Sum (c,f))) is V25() real ext-real set
c is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() Element of K19(K20(NAT,REAL))
x is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() Element of K19(K20(NAT,REAL))
f is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
f + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
e is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer set
Sum (c,e,f) is V25() real ext-real Element of REAL
Sum (x,e,f) is V25() real ext-real Element of REAL
e + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
Sum (c,(e + 1),f) is V25() real ext-real Element of REAL
Sum (x,(e + 1),f) is V25() real ext-real Element of REAL
N0 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
N0 + 0 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
c . N0 is V25() real ext-real Element of REAL
x . N0 is V25() real ext-real Element of REAL
(f + 1) + 0 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
c . (e + 1) is V25() real ext-real Element of REAL
x . (e + 1) is V25() real ext-real Element of REAL
(Sum (c,e,f)) + (c . (e + 1)) is V25() real ext-real Element of REAL
(x . (e + 1)) + (Sum (x,e,f)) is V25() real ext-real Element of REAL
N0 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
c . N0 is V25() real ext-real Element of REAL
x . N0 is V25() real ext-real Element of REAL
Sum (c,(f + 1),f) is V25() real ext-real Element of REAL
Sum (x,(f + 1),f) is V25() real ext-real Element of REAL
Sum (x,(f + 1)) is V25() real ext-real Element of REAL
Sum (x,f) is V25() real ext-real Element of REAL
(Sum (x,(f + 1))) - (Sum (x,f)) is V25() real ext-real Element of REAL
- (Sum (x,f)) is V25() real ext-real set
(Sum (x,(f + 1))) + (- (Sum (x,f))) is V25() real ext-real set
Partial_Sums x is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() Element of K19(K20(NAT,REAL))
(Partial_Sums x) . (f + 1) is V25() real ext-real Element of REAL
((Partial_Sums x) . (f + 1)) - (Sum (x,f)) is V25() real ext-real Element of REAL
((Partial_Sums x) . (f + 1)) + (- (Sum (x,f))) is V25() real ext-real set
x . (f + 1) is V25() real ext-real Element of REAL
(Partial_Sums x) . f is V25() real ext-real Element of REAL
(x . (f + 1)) + ((Partial_Sums x) . f) is V25() real ext-real Element of REAL
((x . (f + 1)) + ((Partial_Sums x) . f)) - (Sum (x,f)) is V25() real ext-real Element of REAL
((x . (f + 1)) + ((Partial_Sums x) . f)) + (- (Sum (x,f))) is V25() real ext-real set
(x . (f + 1)) + (Sum (x,f)) is V25() real ext-real Element of REAL
((x . (f + 1)) + (Sum (x,f))) - (Sum (x,f)) is V25() real ext-real Element of REAL
((x . (f + 1)) + (Sum (x,f))) + (- (Sum (x,f))) is V25() real ext-real set
(x . (f + 1)) + 0 is V25() real ext-real Element of REAL
Sum (c,(f + 1)) is V25() real ext-real Element of REAL
Sum (c,f) is V25() real ext-real Element of REAL
(Sum (c,(f + 1))) - (Sum (c,f)) is V25() real ext-real Element of REAL
- (Sum (c,f)) is V25() real ext-real set
(Sum (c,(f + 1))) + (- (Sum (c,f))) is V25() real ext-real set
Partial_Sums c is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() Element of K19(K20(NAT,REAL))
(Partial_Sums c) . (f + 1) is V25() real ext-real Element of REAL
((Partial_Sums c) . (f + 1)) - (Sum (c,f)) is V25() real ext-real Element of REAL
((Partial_Sums c) . (f + 1)) + (- (Sum (c,f))) is V25() real ext-real set
c . (f + 1) is V25() real ext-real Element of REAL
(Partial_Sums c) . f is V25() real ext-real Element of REAL
(c . (f + 1)) + ((Partial_Sums c) . f) is V25() real ext-real Element of REAL
((c . (f + 1)) + ((Partial_Sums c) . f)) - (Sum (c,f)) is V25() real ext-real Element of REAL
((c . (f + 1)) + ((Partial_Sums c) . f)) + (- (Sum (c,f))) is V25() real ext-real set
(c . (f + 1)) + (Sum (c,f)) is V25() real ext-real Element of REAL
((c . (f + 1)) + (Sum (c,f))) - (Sum (c,f)) is V25() real ext-real Element of REAL
((c . (f + 1)) + (Sum (c,f))) + (- (Sum (c,f))) is V25() real ext-real set
(c . (f + 1)) + 0 is V25() real ext-real Element of REAL
e is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
Sum (c,e,f) is V25() real ext-real Element of REAL
Sum (x,e,f) is V25() real ext-real Element of REAL
c is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
c / 2 is V25() real ext-real non negative Element of REAL
c * (2 ") is V25() real ext-real non negative set
[/(c / 2)\] is V25() real ext-real integer set
0 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
[/(1 / 2)\] is V25() real ext-real integer set
1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(1 / 2) + 1 is non empty V25() real ext-real positive non negative Element of REAL
1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(c / 2) + 1 is non empty V25() real ext-real positive non negative Element of REAL
2 * ((c / 2) + 1) is V25() real ext-real non negative Element of REAL
2 * c is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 * (c / 2) is V25() real ext-real non negative Element of REAL
2 * 1 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(2 * (c / 2)) + (2 * 1) is V25() real ext-real non negative Element of REAL
(2 * c) - c is V25() real ext-real integer Element of REAL
- c is V25() real ext-real non positive integer set
(2 * c) + (- c) is V25() real ext-real integer set
c is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() Element of K19(K20(NAT,REAL))
c . 0 is V25() real ext-real Element of REAL
x is V25() real ext-real Element of REAL
f is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
e is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
Sum (c,f,e) is V25() real ext-real Element of REAL
f - e is V25() real ext-real integer Element of REAL
- e is V25() real ext-real non positive integer set
f + (- e) is V25() real ext-real integer set
x * (f - e) is V25() real ext-real Element of REAL
e + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
Sum (c,f,(e + 1)) is V25() real ext-real Element of REAL
f - (e + 1) is V25() real ext-real integer Element of REAL
- (e + 1) is V25() real ext-real non positive integer set
f + (- (e + 1)) is V25() real ext-real integer set
x * (f - (e + 1)) is V25() real ext-real Element of REAL
Sum (c,f) is V25() real ext-real Element of REAL
Sum (c,(e + 1)) is V25() real ext-real Element of REAL
(Sum (c,f)) - (Sum (c,(e + 1))) is V25() real ext-real Element of REAL
- (Sum (c,(e + 1))) is V25() real ext-real set
(Sum (c,f)) + (- (Sum (c,(e + 1)))) is V25() real ext-real set
Partial_Sums c is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() Element of K19(K20(NAT,REAL))
(Partial_Sums c) . (e + 1) is V25() real ext-real Element of REAL
(Sum (c,f)) - ((Partial_Sums c) . (e + 1)) is V25() real ext-real Element of REAL
- ((Partial_Sums c) . (e + 1)) is V25() real ext-real set
(Sum (c,f)) + (- ((Partial_Sums c) . (e + 1))) is V25() real ext-real set
(Partial_Sums c) . e is V25() real ext-real Element of REAL
c . (e + 1) is V25() real ext-real Element of REAL
((Partial_Sums c) . e) + (c . (e + 1)) is V25() real ext-real Element of REAL
(Sum (c,f)) - (((Partial_Sums c) . e) + (c . (e + 1))) is V25() real ext-real Element of REAL
- (((Partial_Sums c) . e) + (c . (e + 1))) is V25() real ext-real set
(Sum (c,f)) + (- (((Partial_Sums c) . e) + (c . (e + 1)))) is V25() real ext-real set
(Sum (c,f)) - ((Partial_Sums c) . e) is V25() real ext-real Element of REAL
- ((Partial_Sums c) . e) is V25() real ext-real set
(Sum (c,f)) + (- ((Partial_Sums c) . e)) is V25() real ext-real set
- (c . (e + 1)) is V25() real ext-real Element of REAL
((Sum (c,f)) - ((Partial_Sums c) . e)) + (- (c . (e + 1))) is V25() real ext-real Element of REAL
Sum (c,e) is V25() real ext-real Element of REAL
(Sum (c,f)) - (Sum (c,e)) is V25() real ext-real Element of REAL
- (Sum (c,e)) is V25() real ext-real set
(Sum (c,f)) + (- (Sum (c,e))) is V25() real ext-real set
((Sum (c,f)) - (Sum (c,e))) + (- (c . (e + 1))) is V25() real ext-real Element of REAL
(x * (f - e)) + (- (c . (e + 1))) is V25() real ext-real Element of REAL
- x is V25() real ext-real Element of REAL
(x * (f - e)) + (- x) is V25() real ext-real Element of REAL
Sum (c,0) is V25() real ext-real Element of REAL
Partial_Sums c is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() Element of K19(K20(NAT,REAL))
(Partial_Sums c) . 0 is V25() real ext-real Element of REAL
Sum (c,f,0) is V25() real ext-real Element of REAL
Sum (c,f) is V25() real ext-real Element of REAL
(Sum (c,f)) - 0 is V25() real ext-real Element of REAL
- 0 is functional empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V25() real ext-real non positive non negative integer V49() V50() V51() V52() V53() V54() V55() FinSequence-membered set
(Sum (c,f)) + (- 0) is V25() real ext-real set
f - 0 is V25() real ext-real non negative integer Element of REAL
f + (- 0) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer set
x * (f - 0) is V25() real ext-real Element of REAL
e is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
Sum (c,f,e) is V25() real ext-real Element of REAL
f - e is V25() real ext-real integer Element of REAL
- e is V25() real ext-real non positive integer set
f + (- e) is V25() real ext-real integer set
x * (f - e) is V25() real ext-real Element of REAL
c is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() Element of K19(K20(NAT,REAL))
x is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(x) is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() eventually-nonnegative eventually-positive eventually-nonzero Element of K19(K20(NAT,REAL))
x + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
((x + 1)) is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() eventually-nonnegative eventually-positive eventually-nonzero Element of K19(K20(NAT,REAL))
Big_Theta ((x + 1)) is functional non empty FUNCTION_DOMAIN of NAT , REAL
Big_Oh ((x + 1)) is functional non empty FUNCTION_DOMAIN of NAT , REAL
{ b₁ where b₁ is V1() V4( NAT ) V5( REAL ) Function-like V14( NAT ) quasi_total V35() V36() V37() Element of Funcs (NAT,REAL) : ex b₂ being V25() real ext-real Element of REAL ex b₃ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT st
( not b₂ <= 0 & ( for b₄ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT holds
( not b₃ <= b₄ or ( b₁ . b₄ <= b₂ * (((x + 1)) . b₄) & 0 <= b₁ . b₄ ) ) ) ) } is set
Big_Omega ((x + 1)) is functional non empty FUNCTION_DOMAIN of NAT , REAL
{ b₁ where b₁ is V1() V4( NAT ) V5( REAL ) Function-like V14( NAT ) quasi_total V35() V36() V37() Element of Funcs (NAT,REAL) : ex b₂ being V25() real ext-real Element of REAL ex b₃ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT st
( not b₂ <= 0 & ( for b₄ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT holds
( not b₃ <= b₄ or ( b₂ * (((x + 1)) . b₄) <= b₁ . b₄ & 0 <= b₁ . b₄ ) ) ) ) } is set
(Big_Oh ((x + 1))) /\ (Big_Omega ((x + 1))) is set
N0 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
N0 / 2 is V25() real ext-real non negative Element of REAL
N0 * (2 ") is V25() real ext-real non negative set
[/(N0 / 2)\] is V25() real ext-real integer set
(N0 / 2) to_power x is V25() real ext-real Element of REAL
(N0 / 2) |^ x is set
d is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
d is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() Element of K19(K20(NAT,REAL))
d . 0 is V25() real ext-real Element of REAL
N1 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
d . N1 is V25() real ext-real Element of REAL
d is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() Element of K19(K20(NAT,REAL))
d . 0 is V25() real ext-real Element of REAL
d is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() Element of K19(K20(NAT,REAL))
d . 0 is V25() real ext-real Element of REAL
N1 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
N1 - 1 is V25() real ext-real integer Element of REAL
- 1 is V25() real ext-real non positive integer set
N1 + (- 1) is V25() real ext-real integer set
2 " is non empty V25() real ext-real positive non negative Element of REAL
N0 * (2 ") is V25() real ext-real non negative Element of REAL
0 * (2 ") is functional empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V25() real ext-real non positive non negative integer V49() V50() V51() V52() V53() V54() V55() FinSequence-membered Element of REAL
(N1 - 1) + 1 is V25() real ext-real integer Element of REAL
(- 1) + 1 is V25() real ext-real integer Element of REAL
(N0 / 2) + 1 is non empty V25() real ext-real positive non negative Element of REAL
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
(N0 / 2) + (N0 / 2) is V25() real ext-real non negative Element of REAL
(N0 / 2) + N is V25() real ext-real non negative Element of REAL
N0 - N is V25() real ext-real integer Element of REAL
- N is V25() real ext-real non positive integer set
N0 + (- N) is V25() real ext-real integer set
Sum (d,N0,N) is V25() real ext-real Element of REAL
(N0 - N) * ((N0 / 2) to_power x) is V25() real ext-real Element of REAL
(N0 / 2) * ((N0 / 2) to_power x) is V25() real ext-real Element of REAL
(N0 / 2) to_power 1 is V25() real ext-real Element of REAL
(N0 / 2) |^ 1 is set
((N0 / 2) to_power 1) * ((N0 / 2) to_power x) is V25() real ext-real Element of REAL
(N0 / 2) to_power (x + 1) is V25() real ext-real Element of REAL
(N0 / 2) |^ (x + 1) is set
N0 to_power (x + 1) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
N0 |^ (x + 1) is set
2 to_power (x + 1) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 |^ (x + 1) is set
(N0 to_power (x + 1)) / (2 to_power (x + 1)) is V25() real ext-real non negative Element of REAL
(2 to_power (x + 1)) " is V25() real ext-real non negative set
(N0 to_power (x + 1)) * ((2 to_power (x + 1)) ") is V25() real ext-real non negative set
c . N0 is V25() real ext-real Element of REAL
Sum ((x),N0) is V25() real ext-real Element of REAL
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
(x) . n is V25() real ext-real Element of REAL
(x) . n is V25() real ext-real Element of REAL
n to_power x is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
n |^ x is set
N1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
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
(x) . n is V25() real ext-real Element of REAL
Sum ((x),N) is V25() real ext-real Element of REAL
(Sum ((x),N0)) + 0 is V25() real ext-real Element of REAL
(Sum ((x),N0)) + (Sum ((x),N)) is V25() real ext-real Element of REAL
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
d . n is V25() real ext-real Element of REAL
(x) . n is V25() real ext-real Element of REAL
n to_power x is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
n |^ x is set
N + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
Sum ((x),N0,N) is V25() real ext-real Element of REAL
(2 to_power (x + 1)) " is V25() real ext-real non negative Element of REAL
(N0 to_power (x + 1)) * ((2 to_power (x + 1)) ") is V25() real ext-real non negative Element of REAL
(Sum ((x),N0)) - (Sum ((x),N)) is V25() real ext-real Element of REAL
- (Sum ((x),N)) is V25() real ext-real set
(Sum ((x),N0)) + (- (Sum ((x),N))) is V25() real ext-real set
((x + 1)) . N0 is V25() real ext-real Element of REAL
((2 to_power (x + 1)) ") * (((x + 1)) . N0) is V25() real ext-real Element of REAL
N0 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
N0 to_power x is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
N0 |^ x is set
t is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
t is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() Element of K19(K20(NAT,REAL))
t . 0 is V25() real ext-real Element of REAL
d is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
t . d is V25() real ext-real Element of REAL
t is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() Element of K19(K20(NAT,REAL))
t . 0 is V25() real ext-real Element of REAL
t is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() Element of K19(K20(NAT,REAL))
t . 0 is V25() real ext-real Element of REAL
d is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(x) . d is V25() real ext-real Element of REAL
t . d is V25() real ext-real Element of REAL
t . d is V25() real ext-real Element of REAL
(x) . d is V25() real ext-real Element of REAL
d to_power x is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
d |^ x is set
Sum ((x),N0) is V25() real ext-real Element of REAL
Sum (t,N0) is V25() real ext-real Element of REAL
(N0 to_power x) * N0 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
N0 to_power 1 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
N0 |^ 1 is set
(N0 to_power x) * (N0 to_power 1) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
N0 to_power (x + 1) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
N0 |^ (x + 1) is set
((x + 1)) . N0 is V25() real ext-real Element of REAL
c . N0 is V25() real ext-real Element of REAL
1 * (((x + 1)) . N0) is V25() real ext-real Element of REAL
d is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(x) . d is V25() real ext-real Element of REAL
(x) . d is V25() real ext-real Element of REAL
d to_power x is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
d |^ x is set
c is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() Element of K19(K20(NAT,REAL))
x is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
c . x is V25() real ext-real Element of REAL
log (2,x) is V25() real ext-real Element of REAL
x to_power (log (2,x)) is V25() real ext-real Element of REAL
c taken_every 2 is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() Element of K19(K20(NAT,REAL))
f is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() eventually-nonnegative eventually-positive eventually-nonzero Element of K19(K20(NAT,REAL))
Big_Oh f is functional non empty FUNCTION_DOMAIN of NAT , REAL
{ b₁ where b₁ is V1() V4( NAT ) V5( REAL ) Function-like V14( NAT ) quasi_total V35() V36() V37() Element of Funcs (NAT,REAL) : ex b₂ being V25() real ext-real Element of REAL ex b₃ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT st
( not b₂ <= 0 & ( for b₄ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT holds
( not b₃ <= b₄ or ( b₁ . b₄ <= b₂ * (f . b₄) & 0 <= b₁ . b₄ ) ) ) ) } is set
e is V1() V4( NAT ) V5( REAL ) Function-like V14( NAT ) quasi_total V35() V36() V37() Element of Funcs (NAT,REAL)
N0 is V25() real ext-real Element of REAL
t is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
N0 is V25() real ext-real Element of REAL
t is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
sqrt N0 is V25() real ext-real Element of REAL
sqrt 2 is V25() real ext-real Element of REAL
(sqrt N0) / (sqrt 2) is V25() real ext-real Element of REAL
(sqrt 2) " is V25() real ext-real set
(sqrt N0) * ((sqrt 2) ") is V25() real ext-real set
[/((sqrt N0) / (sqrt 2))\] is V25() real ext-real integer set
max (t,[/((sqrt N0) / (sqrt 2))\]) is V25() real ext-real set
N1 is V25() real ext-real integer set
max (N1,2) is V25() real ext-real set
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
N + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
log (2,(N + 1)) is V25() real ext-real Element of REAL
(N + 1) to_power (log (2,(N + 1))) is V25() real ext-real Element of REAL
2 * (N + 1) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 * 0 is 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
N + 0 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(N + 1) * (sqrt 2) is V25() real ext-real Element of REAL
((sqrt N0) / (sqrt 2)) * (sqrt 2) is V25() real ext-real Element of REAL
((N + 1) * (sqrt 2)) ^2 is V25() real ext-real Element of REAL
((N + 1) * (sqrt 2)) * ((N + 1) * (sqrt 2)) is V25() real ext-real set
(sqrt N0) ^2 is V25() real ext-real Element of REAL
(sqrt N0) * (sqrt N0) is V25() real ext-real set
(N + 1) ^2 is V25() real ext-real Element of REAL
(N + 1) * (N + 1) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer set
(sqrt 2) ^2 is V25() real ext-real Element of REAL
(sqrt 2) * (sqrt 2) is V25() real ext-real set
((N + 1) ^2) * ((sqrt 2) ^2) is V25() real ext-real Element of REAL
2 * ((N + 1) ^2) is V25() real ext-real Element of REAL
(2 * ((N + 1) ^2)) * ((N + 1) to_power (log (2,(N + 1)))) is V25() real ext-real Element of REAL
(2 * (N + 1)) * (N + 1) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
((2 * (N + 1)) * (N + 1)) * ((N + 1) to_power (log (2,(N + 1)))) is V25() real ext-real Element of REAL
2 to_power (log (2,(N + 1))) is V25() real ext-real Element of REAL
(2 * (N + 1)) * (2 to_power (log (2,(N + 1)))) is V25() real ext-real Element of REAL
((2 * (N + 1)) * (2 to_power (log (2,(N + 1))))) * ((N + 1) to_power (log (2,(N + 1)))) is V25() real ext-real Element of REAL
(2 to_power (log (2,(N + 1)))) * ((N + 1) to_power (log (2,(N + 1)))) is V25() real ext-real Element of REAL
(2 * (N + 1)) * ((2 to_power (log (2,(N + 1)))) * ((N + 1) to_power (log (2,(N + 1))))) is V25() real ext-real Element of REAL
(2 * (N + 1)) to_power (log (2,(N + 1))) is V25() real ext-real Element of REAL
(2 * (N + 1)) * ((2 * (N + 1)) to_power (log (2,(N + 1)))) is V25() real ext-real Element of REAL
(2 * (N + 1)) to_power 1 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(2 * (N + 1)) |^ 1 is set
((2 * (N + 1)) to_power 1) * ((2 * (N + 1)) to_power (log (2,(N + 1)))) is V25() real ext-real Element of REAL
1 + (log (2,(N + 1))) is V25() real ext-real Element of REAL
(2 * (N + 1)) to_power (1 + (log (2,(N + 1)))) is V25() real ext-real Element of REAL
log (2,2) is V25() real ext-real Element of REAL
(log (2,2)) + (log (2,(N + 1))) is V25() real ext-real Element of REAL
(2 * (N + 1)) to_power ((log (2,2)) + (log (2,(N + 1)))) is V25() real ext-real Element of REAL
log (2,(2 * (N + 1))) is V25() real ext-real Element of REAL
(2 * (N + 1)) to_power (log (2,(2 * (N + 1)))) is V25() real ext-real Element of REAL
N0 * ((N + 1) to_power (log (2,(N + 1)))) is V25() real ext-real Element of REAL
f . (2 * (N + 1)) is V25() real ext-real Element of REAL
e . (N + 1) is V25() real ext-real Element of REAL
f . (N + 1) is V25() real ext-real Element of REAL
N0 * (f . (N + 1)) is V25() real ext-real Element of REAL
c is V25() real ext-real Element of REAL
NAT --> c is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) T-Sequence-like quasi_total V35() V36() V37() Element of K19(K20(NAT,REAL))
(1) is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() Element of K19(K20(NAT,REAL))
NAT --> 1 is V1() V4( NAT ) V5( REAL ) V5( RAT ) V5( INT ) Function-like non empty V14( NAT ) T-Sequence-like quasi_total V35() V36() V37() V38() Element of K19(K20(NAT,REAL))
c is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(1) . c is V25() real ext-real Element of REAL
c is V25() real ext-real Element of REAL
x is V25() real ext-real Element of REAL
f is V25() real ext-real Element of REAL
log (x,f) is V25() real ext-real Element of REAL
log (c,f) is V25() real ext-real Element of REAL
log (x,1) is V25() real ext-real Element of REAL
log (c,c) is V25() real ext-real Element of REAL
log (c,x) is V25() real ext-real Element of REAL
1 * (log (x,f)) is V25() real ext-real Element of REAL
(log (c,x)) * (log (x,f)) is V25() real ext-real Element of REAL
(1) is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() eventually-nonnegative eventually-positive eventually-nonzero Element of K19(K20(NAT,REAL))
{(1)} is non empty set
Big_Oh (1) is functional non empty FUNCTION_DOMAIN of NAT , REAL
{ b₁ where b₁ is V1() V4( NAT ) V5( REAL ) Function-like V14( NAT ) quasi_total V35() V36() V37() Element of Funcs (NAT,REAL) : ex b₂ being V25() real ext-real Element of REAL ex b₃ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT st
( not b₂ <= 0 & ( for b₄ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT holds
( not b₃ <= b₄ or ( b₁ . b₄ <= b₂ * ((1) . b₄) & 0 <= b₁ . b₄ ) ) ) ) } is set
e is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() eventually-nonnegative Element of K19(K20(NAT,REAL))
f is functional non empty FUNCTION_DOMAIN of NAT , REAL
f to_power (Big_Oh (1)) is functional non empty FUNCTION_DOMAIN of NAT , REAL
{ b₁ where b₁ is V1() V4( NAT ) V5( REAL ) Function-like V14( NAT ) quasi_total V35() V36() V37() Element of Funcs (NAT,REAL) : ex b₂, b₃ being V1() V4( NAT ) V5( REAL ) Function-like V14( NAT ) quasi_total V35() V36() V37() Element of Funcs (NAT,REAL) ex b₄ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT st
( b₂ in f & b₃ in Big_Oh (1) & ( for b₅ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT holds
( not b₄ <= b₅ or b₁ . b₅ = (b₂ . b₅) to_power (b₃ . b₅) ) ) ) } is set
t is V1() V4( NAT ) V5( REAL ) Function-like V14( NAT ) quasi_total V35() V36() V37() Element of Funcs (NAT,REAL)
d is V1() V4( NAT ) V5( REAL ) Function-like V14( NAT ) quasi_total V35() V36() V37() Element of Funcs (NAT,REAL)
N1 is V1() V4( NAT ) V5( REAL ) Function-like V14( NAT ) quasi_total V35() V36() V37() Element of Funcs (NAT,REAL)
b is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
d is V1() V4( NAT ) V5( REAL ) Function-like V14( NAT ) quasi_total V35() V36() V37() Element of Funcs (NAT,REAL)
N1 is V1() V4( NAT ) V5( REAL ) Function-like V14( NAT ) quasi_total V35() V36() V37() Element of Funcs (NAT,REAL)
b is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
N is V1() V4( NAT ) V5( REAL ) Function-like V14( NAT ) quasi_total V35() V36() V37() Element of Funcs (NAT,REAL)
n is V25() real ext-real Element of REAL
g9 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
n is V25() real ext-real Element of REAL
g9 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
[/n\] is V25() real ext-real integer set
max (b,g9) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
max (2,(max (b,g9))) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
N0 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
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
c is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
d is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
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
n to_power n is V25() real ext-real Element of REAL
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
n to_power 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
n |^ N is set
d . n is V25() real ext-real Element of REAL
n to_power 1 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
n |^ 1 is set
e . n is V25() real ext-real Element of REAL
N1 . n is V25() real ext-real Element of REAL
n to_power (N1 . n) is V25() real ext-real Element of REAL
N . n is V25() real ext-real Element of REAL
n to_power 0 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
n |^ 0 is set
(1) . n is V25() real ext-real Element of REAL
n * ((1) . n) is V25() real ext-real Element of REAL
n * 1 is V25() real ext-real Element of REAL
n to_power (n * 1) is V25() real ext-real Element of REAL
(N) is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() eventually-nonnegative eventually-positive eventually-nonzero Element of K19(K20(NAT,REAL))
(N) . n is V25() real ext-real Element of REAL
c * ((N) . n) is V25() real ext-real Element of REAL
N1 is V25() real ext-real Element of REAL
d is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
b is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(b) is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() eventually-nonnegative eventually-positive eventually-nonzero Element of K19(K20(NAT,REAL))
max (d,2) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
t is V1() V4( NAT ) V5( REAL ) Function-like V14( NAT ) quasi_total V35() V36() V37() Element of Funcs (NAT,REAL)
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
t . n is V25() real ext-real Element of REAL
log (n,(t . n)) is V25() real ext-real Element of REAL
n is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() Element of K19(K20(NAT,REAL))
g9 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
t . g9 is V25() real ext-real Element of REAL
N0 is V1() V4( NAT ) V5( REAL ) Function-like V14( NAT ) quasi_total V35() V36() V37() Element of Funcs (NAT,REAL)
N0 . g9 is V25() real ext-real Element of REAL
n . g9 is V25() real ext-real Element of REAL
(N0 . g9) to_power (n . g9) is V25() real ext-real Element of REAL
g9 to_power 1 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
g9 |^ 1 is set
(g9 to_power 1) to_power (n . g9) is V25() real ext-real Element of REAL
g9 to_power (n . g9) is V25() real ext-real Element of REAL
log (g9,(t . g9)) is V25() real ext-real Element of REAL
1 * (log (g9,(t . g9))) is V25() real ext-real Element of REAL
g9 to_power (1 * (log (g9,(t . g9)))) is V25() real ext-real Element of REAL
max (N1,2) is V25() real ext-real Element of REAL
log ((max (d,2)),(max (N1,2))) is V25() real ext-real Element of REAL
b + (log ((max (d,2)),(max (N1,2)))) is V25() real ext-real Element of REAL
log ((max (d,2)),1) is V25() real ext-real Element of REAL
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
(1) . n is V25() real ext-real Element of REAL
(b) . n is V25() real ext-real Element of REAL
n to_power b is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
n |^ b is set
N1 * ((b) . n) is V25() real ext-real Element of REAL
(max (N1,2)) * ((b) . n) is V25() real ext-real Element of REAL
log (n,((max (N1,2)) * ((b) . n))) is V25() real ext-real Element of REAL
log (n,(max (N1,2))) is V25() real ext-real Element of REAL
log (n,(n to_power b)) is V25() real ext-real Element of REAL
(log (n,(max (N1,2)))) + (log (n,(n to_power b))) is V25() real ext-real Element of REAL
log (n,n) is V25() real ext-real Element of REAL
b * (log (n,n)) is V25() real ext-real Element of REAL
(log (n,(max (N1,2)))) + (b * (log (n,n))) is V25() real ext-real Element of REAL
b * 1 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(log (n,(max (N1,2)))) + (b * 1) is V25() real ext-real Element of REAL
(log (n,(max (N1,2)))) + b is V25() real ext-real Element of REAL
(log ((max (d,2)),(max (N1,2)))) + b is V25() real ext-real Element of REAL
t . n is V25() real ext-real Element of REAL
N0 . n is V25() real ext-real Element of REAL
n . n is V25() real ext-real Element of REAL
(N0 . n) to_power (n . n) is V25() real ext-real Element of REAL
n to_power 1 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
n |^ 1 is set
(n to_power 1) to_power (n . n) is V25() real ext-real Element of REAL
n to_power (n . n) is V25() real ext-real Element of REAL
log (n,(t . n)) is V25() real ext-real Element of REAL
(n . n) * (log (n,n)) is V25() real ext-real Element of REAL
(n . n) * 1 is V25() real ext-real Element of REAL
(b + (log ((max (d,2)),(max (N1,2))))) * ((1) . n) is V25() real ext-real Element of REAL
log (n,1) is V25() real ext-real Element of REAL
t is V25() real ext-real Element of REAL
N0 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
d is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(d) is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() eventually-nonnegative eventually-positive eventually-nonzero Element of K19(K20(NAT,REAL))
b is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
N1 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
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
(N) is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() eventually-nonnegative eventually-positive eventually-nonzero Element of K19(K20(NAT,REAL))
10 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
10 to_power 6 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
10 |^ 6 is set
3 * (10 to_power 6) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
18 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
10 to_power 3 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
10 |^ 3 is set
18 * (10 to_power 3) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
27 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(2) is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() eventually-nonnegative eventually-positive eventually-nonzero Element of K19(K20(NAT,REAL))
Big_Oh (2) is functional non empty FUNCTION_DOMAIN of NAT , REAL
{ b₁ where b₁ is V1() V4( NAT ) V5( REAL ) Function-like V14( NAT ) quasi_total V35() V36() V37() Element of Funcs (NAT,REAL) : ex b₂ being V25() real ext-real Element of REAL ex b₃ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT st
( not b₂ <= 0 & ( for b₄ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT holds
( not b₃ <= b₄ or ( b₁ . b₄ <= b₂ * ((2) . b₄) & 0 <= b₁ . b₄ ) ) ) ) } is set
10 * 10 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(10 * 10) * 10 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
x is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
x * x is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
f is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
10 ^2 is V25() real ext-real Element of REAL
10 * 10 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer set
10 * (10 ^2) is V25() real ext-real Element of REAL
10 to_power 2 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
10 |^ 2 is set
10 * (10 to_power 2) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
10 to_power 1 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
10 |^ 1 is set
(10 to_power 1) * (10 to_power 2) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
1 + 2 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
10 to_power (1 + 2) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
10 |^ (1 + 2) is set
3 + 3 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
10 to_power (3 + 3) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
10 |^ (3 + 3) is set
2 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
10 to_power (2 + 1) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
10 |^ (2 + 1) is set
(10 to_power 2) * (10 to_power 1) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(10 to_power 2) * 10 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(10 ^2) * 10 is V25() real ext-real Element of REAL
1000 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
400 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
18 * x is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
3 * f is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
e is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer set
e ^2 is V25() real ext-real set
e * e is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer set
27 * (e ^2) is V25() real ext-real Element of REAL
(18 * x) * e is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
((18 * x) * e) - (3 * f) is V25() real ext-real integer Element of REAL
- (3 * f) is V25() real ext-real non positive integer set
((18 * x) * e) + (- (3 * f)) is V25() real ext-real integer set
e + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(e + 1) ^2 is V25() real ext-real set
(e + 1) * (e + 1) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer set
27 * ((e + 1) ^2) is V25() real ext-real Element of REAL
(18 * x) * (e + 1) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
((18 * x) * (e + 1)) - (3 * f) is V25() real ext-real integer Element of REAL
((18 * x) * (e + 1)) + (- (3 * f)) is V25() real ext-real integer set
54 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
54 * 400 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
54 * e is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(54 * e) + 0 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(54 * e) + 27 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(27 * (e ^2)) + ((54 * e) + 27) is V25() real ext-real Element of REAL
(27 * (e ^2)) + (18 * x) is V25() real ext-real Element of REAL
(((18 * x) * e) - (3 * f)) + (18 * x) is V25() real ext-real integer Element of REAL
400 ^2 is V25() real ext-real set
400 * 400 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer set
27 * (400 ^2) is V25() real ext-real Element of REAL
(18 * x) * 400 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
((18 * x) * 400) - (3 * f) is V25() real ext-real integer Element of REAL
- (3 * f) is V25() real ext-real non positive integer set
((18 * x) * 400) + (- (3 * f)) is V25() real ext-real integer set
e is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
e ^2 is V25() real ext-real Element of REAL
e * e is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer set
27 * (e ^2) is V25() real ext-real Element of REAL
(18 * x) * e is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
((18 * x) * e) - (3 * f) is V25() real ext-real integer Element of REAL
((18 * x) * e) + (- (3 * f)) is V25() real ext-real integer set
e is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() Element of K19(K20(NAT,REAL))
N0 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
e . N0 is V25() real ext-real Element of REAL
N0 ^2 is V25() real ext-real Element of REAL
N0 * N0 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer set
27 * (N0 ^2) is V25() real ext-real Element of REAL
(18 * (10 to_power 3)) * N0 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(3 * f) - ((18 * (10 to_power 3)) * N0) is V25() real ext-real integer Element of REAL
- ((18 * (10 to_power 3)) * N0) is V25() real ext-real non positive integer set
(3 * f) + (- ((18 * (10 to_power 3)) * N0)) is V25() real ext-real integer set
((3 * f) - ((18 * (10 to_power 3)) * N0)) + (27 * (N0 ^2)) is V25() real ext-real Element of REAL
(18 * x) * N0 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
- ((18 * x) * N0) is V25() real ext-real non positive integer Element of REAL
(3 * f) + (- ((18 * x) * N0)) is V25() real ext-real integer Element of REAL
(27 * (N0 ^2)) - (27 * (N0 ^2)) is V25() real ext-real Element of REAL
- (27 * (N0 ^2)) is V25() real ext-real set
(27 * (N0 ^2)) + (- (27 * (N0 ^2))) is V25() real ext-real set
(3 * f) - ((18 * x) * N0) is V25() real ext-real integer Element of REAL
- ((18 * x) * N0) is V25() real ext-real non positive integer set
(3 * f) + (- ((18 * x) * N0)) is V25() real ext-real integer set
0 + ((18 * x) * N0) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(18 * x) * 400 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
3 * (10 to_power (3 + 3)) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
7200 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
x * 7200 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(10 to_power 3) * (10 to_power 3) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
3 * ((10 to_power 3) * (10 to_power 3)) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
N0 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
e . N0 is V25() real ext-real Element of REAL
N0 ^2 is V25() real ext-real Element of REAL
N0 * N0 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer set
27 * (N0 ^2) is V25() real ext-real Element of REAL
N0 to_power 2 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
N0 |^ 2 is set
27 * (N0 to_power 2) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(2) . N0 is V25() real ext-real Element of REAL
27 * ((2) . N0) is V25() real ext-real Element of REAL
(18 * x) * N0 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
((18 * x) * N0) - (3 * f) is V25() real ext-real integer Element of REAL
- (3 * f) is V25() real ext-real non positive integer set
((18 * x) * N0) + (- (3 * f)) is V25() real ext-real integer set
0 + (((18 * x) * N0) - (3 * f)) is V25() real ext-real integer Element of REAL
(27 * (N0 ^2)) - (((18 * x) * N0) - (3 * f)) is V25() real ext-real Element of REAL
- (((18 * x) * N0) - (3 * f)) is V25() real ext-real integer set
(27 * (N0 ^2)) + (- (((18 * x) * N0) - (3 * f))) is V25() real ext-real set
(3 * (10 to_power 6)) - ((18 * x) * N0) is V25() real ext-real integer Element of REAL
- ((18 * x) * N0) is V25() real ext-real non positive integer set
(3 * (10 to_power 6)) + (- ((18 * x) * N0)) is V25() real ext-real integer set
((3 * (10 to_power 6)) - ((18 * x) * N0)) + (27 * (N0 ^2)) is V25() real ext-real Element of REAL
(3) is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() eventually-nonnegative eventually-positive eventually-nonzero Element of K19(K20(NAT,REAL))
Big_Oh (3) is functional non empty FUNCTION_DOMAIN of NAT , REAL
{ b₁ where b₁ is V1() V4( NAT ) V5( REAL ) Function-like V14( NAT ) quasi_total V35() V36() V37() Element of Funcs (NAT,REAL) : ex b₂ being V25() real ext-real Element of REAL ex b₃ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT st
( not b₂ <= 0 & ( for b₄ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT holds
( not b₃ <= b₄ or ( b₁ . b₄ <= b₂ * ((3) . b₄) & 0 <= b₁ . b₄ ) ) ) ) } is set
f is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(2) . f is V25() real ext-real Element of REAL
f to_power 2 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
f |^ 2 is set
(3) . f is V25() real ext-real Element of REAL
f to_power 3 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
f |^ 3 is set
1 * ((3) . f) is V25() real ext-real Element of REAL
Big_Omega (3) is functional non empty FUNCTION_DOMAIN of NAT , REAL
{ b₁ where b₁ is V1() V4( NAT ) V5( REAL ) Function-like V14( NAT ) quasi_total V35() V36() V37() Element of Funcs (NAT,REAL) : ex b₂ being V25() real ext-real Element of REAL ex b₃ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT st
( not b₂ <= 0 & ( for b₄ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT holds
( not b₃ <= b₄ or ( b₂ * ((3) . b₄) <= b₁ . b₄ & 0 <= b₁ . b₄ ) ) ) ) } is set
f is V1() V4( NAT ) V5( REAL ) Function-like V14( NAT ) quasi_total V35() V36() V37() Element of Funcs (NAT,REAL)
e is V25() real ext-real Element of REAL
N0 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
e is V25() real ext-real Element of REAL
N0 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
N0 + 2 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
1 + 0 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(N0 + 2) / e is V25() real ext-real Element of REAL
e " is V25() real ext-real set
(N0 + 2) * (e ") is V25() real ext-real set
[/((N0 + 2) / e)\] is V25() real ext-real integer set
max (N0,[/((N0 + 2) / e)\]) is V25() real ext-real set
t is V25() real ext-real set
(3) . t is V25() real ext-real Element of REAL
e * ((3) . t) is V25() real ext-real Element of REAL
f . t is V25() real ext-real Element of REAL
e " is V25() real ext-real Element of REAL
(N0 + 2) * (e ") is V25() real ext-real Element of REAL
0 * (e ") is V25() real ext-real Element of REAL
d is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(2) . d is V25() real ext-real Element of REAL
- 2 is V25() real ext-real non positive integer Element of REAL
d to_power (- 2) is V25() real ext-real Element of REAL
((2) . d) * (d to_power (- 2)) is V25() real ext-real Element of REAL
d to_power 2 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
d |^ 2 is set
(d to_power 2) * (d to_power (- 2)) is V25() real ext-real Element of REAL
2 + (- 2) is V25() real ext-real integer Element of REAL
d to_power (2 + (- 2)) is V25() real ext-real Element of REAL
e * [/((N0 + 2) / e)\] is V25() real ext-real Element of REAL
e * d is V25() real ext-real Element of REAL
e * ((N0 + 2) / e) is V25() real ext-real Element of REAL
((N0 + 2) / e) * e is V25() real ext-real Element of REAL
(3) . d is V25() real ext-real Element of REAL
e * ((3) . d) is V25() real ext-real Element of REAL
(e * ((3) . d)) * (d to_power (- 2)) is V25() real ext-real Element of REAL
d to_power 3 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
d |^ 3 is set
e * (d to_power 3) is V25() real ext-real Element of REAL
(e * (d to_power 3)) * (d to_power (- 2)) is V25() real ext-real Element of REAL
(d to_power 3) * (d to_power (- 2)) is V25() real ext-real Element of REAL
e * ((d to_power 3) * (d to_power (- 2))) is V25() real ext-real Element of REAL
3 + (- 2) is V25() real ext-real integer Element of REAL
d to_power (3 + (- 2)) is V25() real ext-real Element of REAL
e * (d to_power (3 + (- 2))) is V25() real ext-real Element of REAL
t is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(3) . t is V25() real ext-real Element of REAL
e * ((3) . t) is V25() real ext-real Element of REAL
f . t is V25() real ext-real Element of REAL
(2,1,1) is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() eventually-nonnegative eventually-positive eventually-nonzero Element of K19(K20(NAT,REAL))
(2,1,0) is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() eventually-nonnegative eventually-positive eventually-nonzero Element of K19(K20(NAT,REAL))
c is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() eventually-nonnegative eventually-positive eventually-nonzero Element of K19(K20(NAT,REAL))
Big_Theta c is functional non empty FUNCTION_DOMAIN of NAT , REAL
Big_Oh c is functional non empty FUNCTION_DOMAIN of NAT , REAL
{ b₁ where b₁ is V1() V4( NAT ) V5( REAL ) Function-like V14( NAT ) quasi_total V35() V36() V37() Element of Funcs (NAT,REAL) : ex b₂ being V25() real ext-real Element of REAL ex b₃ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT st
( not b₂ <= 0 & ( for b₄ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT holds
( not b₃ <= b₄ or ( b₁ . b₄ <= b₂ * (c . b₄) & 0 <= b₁ . b₄ ) ) ) ) } is set
Big_Omega c is functional non empty FUNCTION_DOMAIN of NAT , REAL
{ b₁ where b₁ is V1() V4( NAT ) V5( REAL ) Function-like V14( NAT ) quasi_total V35() V36() V37() Element of Funcs (NAT,REAL) : ex b₂ being V25() real ext-real Element of REAL ex b₃ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT st
( not b₂ <= 0 & ( for b₄ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT holds
( not b₃ <= b₄ or ( b₂ * (c . b₄) <= b₁ . b₄ & 0 <= b₁ . b₄ ) ) ) ) } is set
(Big_Oh c) /\ (Big_Omega c) is set
f is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(2,1,0) . f is V25() real ext-real Element of REAL
1 * f is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(1 * f) + 0 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 to_power ((1 * f) + 0) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 |^ ((1 * f) + 0) is set
c . f is V25() real ext-real Element of REAL
(1 * f) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 to_power ((1 * f) + 1) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 |^ ((1 * f) + 1) is set
2 to_power (- 1) is V25() real ext-real Element of REAL
(2 to_power (- 1)) * (c . f) is V25() real ext-real Element of REAL
f + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(- 1) + (f + 1) is V25() real ext-real integer Element of REAL
2 to_power ((- 1) + (f + 1)) is V25() real ext-real Element of REAL
f + 0 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
1 * (c . f) is V25() real ext-real Element of REAL
{ b₁ where b₁ is V1() V4( NAT ) V5( REAL ) Function-like V14( NAT ) quasi_total V35() V36() V37() Element of Funcs (NAT,REAL) : ex b₂, b₃ being V25() real ext-real Element of REAL ex b₄ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT st
( not b₂ <= 0 & not b₃ <= 0 & ( for b₅ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT holds
( not b₄ <= b₅ or ( b₃ * (c . b₅) <= b₁ . b₅ & b₁ . b₅ <= b₂ * (c . b₅) ) ) ) ) } is set
c is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
x is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() Element of K19(K20(NAT,REAL))
f is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
x . f is V25() real ext-real Element of REAL
f + c is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(f + c) ! is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
x is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() Element of K19(K20(NAT,REAL))
f is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() Element of K19(K20(NAT,REAL))
e is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
x . e is V25() real ext-real Element of REAL
e + c is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(e + c) ! is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
f . e is V25() real ext-real Element of REAL
c is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(c) is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() Element of K19(K20(NAT,REAL))
f is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(c) . f is V25() real ext-real Element of REAL
f + c is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(f + c) ! is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(0) is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() eventually-nonnegative eventually-positive eventually-nonzero Element of K19(K20(NAT,REAL))
(1) is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() eventually-nonnegative eventually-positive eventually-nonzero Element of K19(K20(NAT,REAL))
Big_Theta (1) is functional non empty FUNCTION_DOMAIN of NAT , REAL
Big_Oh (1) is functional non empty FUNCTION_DOMAIN of NAT , REAL
{ b₁ where b₁ is V1() V4( NAT ) V5( REAL ) Function-like V14( NAT ) quasi_total V35() V36() V37() Element of Funcs (NAT,REAL) : ex b₂ being V25() real ext-real Element of REAL ex b₃ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT st
( not b₂ <= 0 & ( for b₄ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT holds
( not b₃ <= b₄ or ( b₁ . b₄ <= b₂ * ((1) . b₄) & 0 <= b₁ . b₄ ) ) ) ) } is set
Big_Omega (1) is functional non empty FUNCTION_DOMAIN of NAT , REAL
{ b₁ where b₁ is V1() V4( NAT ) V5( REAL ) Function-like V14( NAT ) quasi_total V35() V36() V37() Element of Funcs (NAT,REAL) : ex b₂ being V25() real ext-real Element of REAL ex b₃ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT st
( not b₂ <= 0 & ( for b₄ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT holds
( not b₃ <= b₄ or ( b₂ * ((1) . b₄) <= b₁ . b₄ & 0 <= b₁ . b₄ ) ) ) ) } is set
(Big_Oh (1)) /\ (Big_Omega (1)) is set
{ b₁ where b₁ is V1() V4( NAT ) V5( REAL ) Function-like V14( NAT ) quasi_total V35() V36() V37() Element of Funcs (NAT,REAL) : ex b₂, b₃ being V25() real ext-real Element of REAL ex b₄ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT st
( not b₂ <= 0 & not b₃ <= 0 & ( for b₅ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT holds
( not b₄ <= b₅ or ( b₃ * ((1) . b₅) <= b₁ . b₅ & b₁ . b₅ <= b₂ * ((1) . b₅) ) ) ) ) } is set
f is V1() V4( NAT ) V5( REAL ) Function-like V14( NAT ) quasi_total V35() V36() V37() Element of Funcs (NAT,REAL)
e is V25() real ext-real Element of REAL
N0 is V25() real ext-real Element of REAL
t is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
e is V25() real ext-real Element of REAL
N0 is V25() real ext-real Element of REAL
t is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
t + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(t + 1) / N0 is V25() real ext-real Element of REAL
N0 " is V25() real ext-real set
(t + 1) * (N0 ") is V25() real ext-real set
[/((t + 1) / N0)\] is V25() real ext-real integer set
((t + 1) / N0) + 1 is V25() real ext-real Element of REAL
[/((t + 1) / N0)\] + 1 is V25() real ext-real integer Element of REAL
N0 * (((t + 1) / N0) + 1) is V25() real ext-real Element of REAL
N0 * ([/((t + 1) / N0)\] + 1) is V25() real ext-real Element of REAL
1 + 0 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
N0 * ((t + 1) / N0) is V25() real ext-real Element of REAL
N0 * 1 is V25() real ext-real Element of REAL
(N0 * ((t + 1) / N0)) + (N0 * 1) is V25() real ext-real Element of REAL
(t + 1) + N0 is V25() real ext-real Element of REAL
max (t,[/((t + 1) / N0)\]) is V25() real ext-real set
d is V25() real ext-real set
(1) . d is V25() real ext-real Element of REAL
N0 * ((1) . d) is V25() real ext-real Element of REAL
(0) . d is V25() real ext-real Element of REAL
N1 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
N1 ! is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
N1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
N0 * (N1 + 1) is V25() real ext-real Element of REAL
(0) . N1 is V25() real ext-real Element of REAL
(N1 !) " is V25() real ext-real non negative Element of REAL
((0) . N1) * ((N1 !) ") is V25() real ext-real Element of REAL
N1 + 0 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(N1 + 0) ! is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
((N1 + 0) !) * ((N1 !) ") is V25() real ext-real non negative Element of REAL
(1) . N1 is V25() real ext-real Element of REAL
N0 * ((1) . N1) is V25() real ext-real Element of REAL
(N0 * ((1) . N1)) * ((N1 !) ") is V25() real ext-real Element of REAL
(N1 + 1) ! is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
N0 * ((N1 + 1) !) is V25() real ext-real Element of REAL
(N0 * ((N1 + 1) !)) * ((N1 !) ") is V25() real ext-real Element of REAL
(N1 + 1) * (N1 !) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
N0 * ((N1 + 1) * (N1 !)) is V25() real ext-real Element of REAL
(N0 * ((N1 + 1) * (N1 !))) * ((N1 !) ") is V25() real ext-real Element of REAL
(N1 !) * ((N1 !) ") is V25() real ext-real non negative Element of REAL
(N0 * (N1 + 1)) * ((N1 !) * ((N1 !) ")) is V25() real ext-real Element of REAL
(N0 * (N1 + 1)) * 1 is V25() real ext-real Element of REAL
d is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(1) . d is V25() real ext-real Element of REAL
N0 * ((1) . d) is V25() real ext-real Element of REAL
(0) . d is V25() real ext-real Element of REAL
x is V25() real ext-real Element of REAL
c is V25() real ext-real Element of REAL
f is V25() real ext-real Element of REAL
e is V25() real ext-real Element of REAL
x * f is V25() real ext-real Element of REAL
x * e is V25() real ext-real Element of REAL
c * f is V25() real ext-real Element of REAL
Big_Oh (1) is functional non empty FUNCTION_DOMAIN of NAT , REAL
{ b₁ where b₁ is V1() V4( NAT ) V5( REAL ) Function-like V14( NAT ) quasi_total V35() V36() V37() Element of Funcs (NAT,REAL) : ex b₂ being V25() real ext-real Element of REAL ex b₃ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT st
( not b₂ <= 0 & ( for b₄ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT holds
( not b₃ <= b₄ or ( b₁ . b₄ <= b₂ * ((1) . b₄) & 0 <= b₁ . b₄ ) ) ) ) } is set
c is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() Element of K19(K20(NAT,REAL))
c (#) c is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() Element of K19(K20(NAT,REAL))
e is V1() V4( NAT ) V5( REAL ) Function-like V14( NAT ) quasi_total V35() V36() V37() Element of Funcs (NAT,REAL)
N0 is V25() real ext-real Element of REAL
t is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
N0 is V25() real ext-real Element of REAL
t is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
N0 * N0 is V25() real ext-real Element of REAL
max (N0,(N0 * N0)) is V25() real ext-real Element of REAL
0 to_power 1 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
0 |^ 1 is set
N1 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
b is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
e . b is V25() real ext-real Element of REAL
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
(1) . N is V25() real ext-real Element of REAL
(2) . N is V25() real ext-real Element of REAL
0 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
1 to_power 2 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
1 |^ 2 is set
1 to_power 1 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
1 |^ 1 is set
N to_power 2 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
N |^ 2 is set
N to_power 1 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
N |^ 1 is set
(1) . b is V25() real ext-real Element of REAL
(2) . b is V25() real ext-real Element of REAL
b to_power 1 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
b |^ 1 is set
(N0 * N0) * ((2) . b) is V25() real ext-real Element of REAL
(max (N0,(N0 * N0))) * ((2) . b) is V25() real ext-real Element of REAL
b to_power 1 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
b |^ 1 is set
(b to_power 1) * (b to_power 1) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
b to_power (1 + 1) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
b |^ (1 + 1) is set
((1) . b) * ((1) . b) is V25() real ext-real Element of REAL
0 * ((1) . b) is V25() real ext-real Element of REAL
(b to_power 1) * ((1) . b) is V25() real ext-real Element of REAL
N0 * ((1) . b) is V25() real ext-real Element of REAL
(e . b) * (e . b) is V25() real ext-real Element of REAL
(N0 * ((1) . b)) * (N0 * ((1) . b)) is V25() real ext-real Element of REAL
e (#) e is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() Element of K19(K20(NAT,REAL))
(e (#) e) . b is V25() real ext-real Element of REAL
(e . b) * 0 is V25() real ext-real Element of REAL
N1 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 (#) (1) is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() eventually-nonnegative Element of K19(K20(NAT,REAL))
(2,2,0) is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() eventually-nonnegative eventually-positive eventually-nonzero Element of K19(K20(NAT,REAL))
c is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() eventually-nonnegative eventually-positive eventually-nonzero Element of K19(K20(NAT,REAL))
Big_Oh c is functional non empty FUNCTION_DOMAIN of NAT , REAL
{ b₁ where b₁ is V1() V4( NAT ) V5( REAL ) Function-like V14( NAT ) quasi_total V35() V36() V37() Element of Funcs (NAT,REAL) : ex b₂ being V25() real ext-real Element of REAL ex b₃ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT st
( not b₂ <= 0 & ( for b₄ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT holds
( not b₃ <= b₄ or ( b₁ . b₄ <= b₂ * (c . b₄) & 0 <= b₁ . b₄ ) ) ) ) } is set
N0 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(2 (#) (1)) . N0 is V25() real ext-real Element of REAL
(1) . N0 is V25() real ext-real Element of REAL
2 * ((1) . N0) is V25() real ext-real Element of REAL
N0 to_power 1 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
N0 |^ 1 is set
2 * 0 is 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
2 * N0 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
N0 is V1() V4( NAT ) V5( REAL ) Function-like V14( NAT ) quasi_total V35() V36() V37() Element of Funcs (NAT,REAL)
t is V25() real ext-real Element of REAL
d is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
t is V25() real ext-real Element of REAL
d is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
log (2,t) is V25() real ext-real Element of REAL
(log (2,t)) + 1 is V25() real ext-real Element of REAL
[/((log (2,t)) + 1)\] is V25() real ext-real integer set
max (d,[/((log (2,t)) + 1)\]) is V25() real ext-real set
N1 is V25() real ext-real set
N0 . N1 is V25() real ext-real Element of REAL
c . N1 is V25() real ext-real Element of REAL
t * (c . N1) is V25() real ext-real Element of REAL
2 to_power [/((log (2,t)) + 1)\] is V25() real ext-real set
b is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 to_power b is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 |^ b is set
- b is V25() real ext-real non positive integer Element of REAL
2 to_power (- b) is V25() real ext-real Element of REAL
2 to_power ((log (2,t)) + 1) is V25() real ext-real Element of REAL
2 to_power (log (2,t)) is V25() real ext-real Element of REAL
2 to_power 1 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 |^ 1 is set
(2 to_power (log (2,t))) * (2 to_power 1) is V25() real ext-real Element of REAL
t * (2 to_power 1) is V25() real ext-real Element of REAL
t * 2 is V25() real ext-real Element of REAL
c . b is V25() real ext-real Element of REAL
t * (c . b) is V25() real ext-real Element of REAL
(t * (c . b)) * (2 to_power (- b)) is V25() real ext-real Element of REAL
1 * b is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(1 * b) + 0 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 to_power ((1 * b) + 0) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 |^ ((1 * b) + 0) is set
t * (2 to_power ((1 * b) + 0)) is V25() real ext-real Element of REAL
(t * (2 to_power ((1 * b) + 0))) * (2 to_power (- b)) is V25() real ext-real Element of REAL
(2 to_power b) * (2 to_power (- b)) is V25() real ext-real Element of REAL
t * ((2 to_power b) * (2 to_power (- b))) is V25() real ext-real Element of REAL
b + (- b) is V25() real ext-real integer Element of REAL
2 to_power (b + (- b)) is V25() real ext-real Element of REAL
t * (2 to_power (b + (- b))) is V25() real ext-real Element of REAL
t * 1 is V25() real ext-real Element of REAL
(2,2,0) . b is V25() real ext-real Element of REAL
((2,2,0) . b) * (2 to_power (- b)) is V25() real ext-real Element of REAL
2 * b is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(2 * b) + 0 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 to_power ((2 * b) + 0) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 |^ ((2 * b) + 0) is set
(2 to_power ((2 * b) + 0)) * (2 to_power (- b)) is V25() real ext-real Element of REAL
(- 1) * b is V25() real ext-real non positive integer Element of REAL
(2 * b) + ((- 1) * b) is V25() real ext-real integer Element of REAL
2 to_power ((2 * b) + ((- 1) * b)) is V25() real ext-real Element of REAL
2 to_power (1 * b) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 |^ (1 * b) is set
N1 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
N0 . N1 is V25() real ext-real Element of REAL
c . N1 is V25() real ext-real Element of REAL
t * (c . N1) is V25() real ext-real Element of REAL
159 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
100 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
159 / 100 is V25() real ext-real non negative Element of REAL
100 " is non empty V25() real ext-real positive non negative set
159 * (100 ") is V25() real ext-real non negative set
log (2,3) is V25() real ext-real Element of REAL
((log (2,3))) is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() eventually-nonnegative eventually-positive eventually-nonzero Element of K19(K20(NAT,REAL))
((159 / 100)) is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() eventually-nonnegative eventually-positive eventually-nonzero Element of K19(K20(NAT,REAL))
Big_Oh ((159 / 100)) is functional non empty FUNCTION_DOMAIN of NAT , REAL
{ b₁ where b₁ is V1() V4( NAT ) V5( REAL ) Function-like V14( NAT ) quasi_total V35() V36() V37() Element of Funcs (NAT,REAL) : ex b₂ being V25() real ext-real Element of REAL ex b₃ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT st
( not b₂ <= 0 & ( for b₄ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT holds
( not b₃ <= b₄ or ( b₁ . b₄ <= b₂ * (((159 / 100)) . b₄) & 0 <= b₁ . b₄ ) ) ) ) } is set
Big_Omega ((159 / 100)) is functional non empty FUNCTION_DOMAIN of NAT , REAL
{ b₁ where b₁ is V1() V4( NAT ) V5( REAL ) Function-like V14( NAT ) quasi_total V35() V36() V37() Element of Funcs (NAT,REAL) : ex b₂ being V25() real ext-real Element of REAL ex b₃ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT st
( not b₂ <= 0 & ( for b₄ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT holds
( not b₃ <= b₄ or ( b₂ * (((159 / 100)) . b₄) <= b₁ . b₄ & 0 <= b₁ . b₄ ) ) ) ) } is set
Big_Theta ((159 / 100)) is functional non empty FUNCTION_DOMAIN of NAT , REAL
(Big_Oh ((159 / 100))) /\ (Big_Omega ((159 / 100))) is set
(159 / 100) - (log (2,3)) is V25() real ext-real Element of REAL
- (log (2,3)) is V25() real ext-real set
(159 / 100) + (- (log (2,3))) is V25() real ext-real set
((log (2,3))) /" ((159 / 100)) is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() Element of K19(K20(NAT,REAL))
((159 / 100)) " is V1() V4( NAT ) Function-like V14( NAT ) V35() V36() V37() set
((log (2,3))) (#) (((159 / 100)) ") is V1() V4( NAT ) Function-like V14( NAT ) V35() V36() V37() set
(log (2,3)) - (log (2,3)) is V25() real ext-real Element of REAL
(log (2,3)) + (- (log (2,3))) is V25() real ext-real set
((159 / 100) - (log (2,3))) / 2 is V25() real ext-real Element of REAL
((159 / 100) - (log (2,3))) * (2 ") is V25() real ext-real set
((159 / 100) - (log (2,3))) * 1 is V25() real ext-real Element of REAL
((159 / 100) - (log (2,3))) * (1 / 2) is V25() real ext-real Element of REAL
N0 is V25() real ext-real set
t is V25() real ext-real Element of REAL
1 / t is V25() real ext-real Element of REAL
t " is V25() real ext-real set
1 * (t ") is V25() real ext-real set
1 / (((159 / 100) - (log (2,3))) / 2) is V25() real ext-real Element of REAL
(((159 / 100) - (log (2,3))) / 2) " is V25() real ext-real set
1 * ((((159 / 100) - (log (2,3))) / 2) ") is V25() real ext-real set
(1 / t) to_power (1 / (((159 / 100) - (log (2,3))) / 2)) is V25() real ext-real Element of REAL
[/((1 / t) to_power (1 / (((159 / 100) - (log (2,3))) / 2)))\] is V25() real ext-real integer set
max ([/((1 / t) to_power (1 / (((159 / 100) - (log (2,3))) / 2)))\],2) is V25() real ext-real set
1 / N0 is V25() real ext-real Element of REAL
N0 " is V25() real ext-real set
1 * (N0 ") is V25() real ext-real set
(1 / N0) to_power (1 / (((159 / 100) - (log (2,3))) / 2)) is V25() real ext-real Element of REAL
[/((1 / N0) to_power (1 / (((159 / 100) - (log (2,3))) / 2)))\] is V25() real ext-real integer set
N1 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
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
(((log (2,3))) /" ((159 / 100))) . N is V25() real ext-real Element of REAL
((log (2,3))) . N is V25() real ext-real Element of REAL
((159 / 100)) . N is V25() real ext-real Element of REAL
(((log (2,3))) . N) / (((159 / 100)) . N) is V25() real ext-real Element of REAL
(((159 / 100)) . N) " is V25() real ext-real set
(((log (2,3))) . N) * ((((159 / 100)) . N) ") is V25() real ext-real set
b is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
N to_power (log (2,3)) is V25() real ext-real Element of REAL
N to_power (159 / 100) is V25() real ext-real Element of REAL
(N to_power (log (2,3))) / (N to_power (159 / 100)) is V25() real ext-real Element of REAL
(N to_power (159 / 100)) " is V25() real ext-real set
(N to_power (log (2,3))) * ((N to_power (159 / 100)) ") is V25() real ext-real set
(log (2,3)) - (159 / 100) is V25() real ext-real Element of REAL
- (159 / 100) is V25() real ext-real non positive set
(log (2,3)) + (- (159 / 100)) is V25() real ext-real set
N to_power ((log (2,3)) - (159 / 100)) is V25() real ext-real Element of REAL
- ((159 / 100) - (log (2,3))) is V25() real ext-real Element of REAL
N to_power (- ((159 / 100) - (log (2,3)))) is V25() real ext-real Element of REAL
((1 / N0) to_power (1 / (((159 / 100) - (log (2,3))) / 2))) to_power (((159 / 100) - (log (2,3))) / 2) is V25() real ext-real Element of REAL
N to_power (((159 / 100) - (log (2,3))) / 2) is V25() real ext-real Element of REAL
(1 / (((159 / 100) - (log (2,3))) / 2)) * (((159 / 100) - (log (2,3))) / 2) is V25() real ext-real Element of REAL
(1 / t) to_power ((1 / (((159 / 100) - (log (2,3))) / 2)) * (((159 / 100) - (log (2,3))) / 2)) is V25() real ext-real Element of REAL
(1 / N0) to_power 1 is V25() real ext-real Element of REAL
(1 / N0) |^ 1 is set
1 / (N to_power (((159 / 100) - (log (2,3))) / 2)) is V25() real ext-real Element of REAL
(N to_power (((159 / 100) - (log (2,3))) / 2)) " is V25() real ext-real set
1 * ((N to_power (((159 / 100) - (log (2,3))) / 2)) ") is V25() real ext-real set
1 / (N0 ") is V25() real ext-real Element of REAL
(N0 ") " is V25() real ext-real set
1 * ((N0 ") ") is V25() real ext-real set
- (((159 / 100) - (log (2,3))) / 2) is V25() real ext-real Element of REAL
N to_power (- (((159 / 100) - (log (2,3))) / 2)) is V25() real ext-real Element of REAL
N to_power ((159 / 100) - (log (2,3))) is V25() real ext-real Element of REAL
1 / (N to_power ((159 / 100) - (log (2,3)))) is V25() real ext-real Element of REAL
(N to_power ((159 / 100) - (log (2,3)))) " is V25() real ext-real set
1 * ((N to_power ((159 / 100) - (log (2,3)))) ") is V25() real ext-real set
((((log (2,3))) /" ((159 / 100))) . N) - 0 is V25() real ext-real Element of REAL
- 0 is functional empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V25() real ext-real non positive non negative integer V49() V50() V51() V52() V53() V54() V55() FinSequence-membered set
((((log (2,3))) /" ((159 / 100))) . N) + (- 0) is V25() real ext-real set
abs (((((log (2,3))) /" ((159 / 100))) . N) - 0) is V25() real ext-real Element of REAL
lim (((log (2,3))) /" ((159 / 100))) is V25() real ext-real Element of REAL
Big_Oh ((log (2,3))) is functional non empty FUNCTION_DOMAIN of NAT , REAL
{ b₁ where b₁ is V1() V4( NAT ) V5( REAL ) Function-like V14( NAT ) quasi_total V35() V36() V37() Element of Funcs (NAT,REAL) : ex b₂ being V25() real ext-real Element of REAL ex b₃ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT st
( not b₂ <= 0 & ( for b₄ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT holds
( not b₃ <= b₄ or ( b₁ . b₄ <= b₂ * (((log (2,3))) . b₄) & 0 <= b₁ . b₄ ) ) ) ) } is set
c is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() Element of K19(K20(NAT,REAL))
x is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() Element of K19(K20(NAT,REAL))
f is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
x . f is V25() real ext-real Element of REAL
f + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(f + 1) mod 2 is V25() real ext-real integer set
e is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
c . e is V25() real ext-real Element of REAL
e mod 2 is V25() real ext-real integer set
f is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() eventually-nonnegative Element of K19(K20(NAT,REAL))
e is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() eventually-nonnegative Element of K19(K20(NAT,REAL))
Big_Oh e is functional non empty FUNCTION_DOMAIN of NAT , REAL
{ b₁ where b₁ is V1() V4( NAT ) V5( REAL ) Function-like V14( NAT ) quasi_total V35() V36() V37() Element of Funcs (NAT,REAL) : ex b₂ being V25() real ext-real Element of REAL ex b₃ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT st
( not b₂ <= 0 & ( for b₄ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT holds
( not b₃ <= b₄ or ( b₁ . b₄ <= b₂ * (e . b₄) & 0 <= b₁ . b₄ ) ) ) ) } is set
N0 is V1() V4( NAT ) V5( REAL ) Function-like V14( NAT ) quasi_total V35() V36() V37() Element of Funcs (NAT,REAL)
t is V25() real ext-real Element of REAL
d is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
t is V25() real ext-real Element of REAL
d is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
d mod 2 is V25() real ext-real integer set
e . d is V25() real ext-real Element of REAL
t * (e . d) is V25() real ext-real Element of REAL
N0 . d is V25() real ext-real Element of REAL
d + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(d + 1) mod 2 is V25() real ext-real integer set
1 mod 2 is V25() real ext-real integer set
0 + (1 mod 2) is V25() real ext-real integer Element of REAL
(0 + (1 mod 2)) mod 2 is V25() real ext-real integer set
0 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(0 + 1) mod 2 is V25() real ext-real integer set
d mod 2 is V25() real ext-real integer set
d + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
e . (d + 1) is V25() real ext-real Element of REAL
(d + 1) mod 2 is V25() real ext-real integer set
1 mod 2 is V25() real ext-real integer set
1 + (1 mod 2) is V25() real ext-real integer Element of REAL
(1 + (1 mod 2)) mod 2 is V25() real ext-real integer set
1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(1 + 1) mod 2 is V25() real ext-real integer set
t * (e . (d + 1)) is V25() real ext-real Element of REAL
N0 . (d + 1) is V25() real ext-real Element of REAL
(d + 1) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
((d + 1) + 1) mod 2 is V25() real ext-real integer set
d + (1 + 1) is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(d + (1 + 1)) mod 2 is V25() real ext-real integer set
2 mod 2 is V25() real ext-real integer set
1 + (2 mod 2) is V25() real ext-real integer Element of REAL
(1 + (2 mod 2)) mod 2 is V25() real ext-real integer set
1 + 0 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(1 + 0) mod 2 is V25() real ext-real integer set
d mod 2 is V25() real ext-real integer set
N1 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
N0 . N1 is V25() real ext-real Element of REAL
e . N1 is V25() real ext-real Element of REAL
t * (e . N1) is V25() real ext-real Element of REAL
Big_Oh f is functional non empty FUNCTION_DOMAIN of NAT , REAL
{ b₁ where b₁ is V1() V4( NAT ) V5( REAL ) Function-like V14( NAT ) quasi_total V35() V36() V37() Element of Funcs (NAT,REAL) : ex b₂ being V25() real ext-real Element of REAL ex b₃ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT st
( not b₂ <= 0 & ( for b₄ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT holds
( not b₃ <= b₄ or ( b₁ . b₄ <= b₂ * (f . b₄) & 0 <= b₁ . b₄ ) ) ) ) } is set
N0 is V1() V4( NAT ) V5( REAL ) Function-like V14( NAT ) quasi_total V35() V36() V37() Element of Funcs (NAT,REAL)
t is V25() real ext-real Element of REAL
d is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
t is V25() real ext-real Element of REAL
d is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
d mod 2 is V25() real ext-real integer set
d + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
f . (d + 1) is V25() real ext-real Element of REAL
(d + 1) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
((d + 1) + 1) mod 2 is V25() real ext-real integer set
1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
d + (1 + 1) is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(d + (1 + 1)) mod 2 is V25() real ext-real integer set
2 mod 2 is V25() real ext-real integer set
0 + (2 mod 2) is V25() real ext-real integer Element of REAL
(0 + (2 mod 2)) mod 2 is V25() real ext-real integer set
0 + 0 is 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
(0 + 0) mod 2 is V25() real ext-real integer set
t * (f . (d + 1)) is V25() real ext-real Element of REAL
N0 . (d + 1) is V25() real ext-real Element of REAL
(d + 1) mod 2 is V25() real ext-real integer set
1 mod 2 is V25() real ext-real integer set
0 + (1 mod 2) is V25() real ext-real integer Element of REAL
(0 + (1 mod 2)) mod 2 is V25() real ext-real integer set
0 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(0 + 1) mod 2 is V25() real ext-real integer set
d mod 2 is V25() real ext-real integer set
f . d is V25() real ext-real Element of REAL
d + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(d + 1) mod 2 is V25() real ext-real integer set
1 mod 2 is V25() real ext-real integer set
1 + (1 mod 2) is V25() real ext-real integer Element of REAL
(1 + (1 mod 2)) mod 2 is V25() real ext-real integer set
1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(1 + 1) mod 2 is V25() real ext-real integer set
t * (f . d) is V25() real ext-real Element of REAL
N0 . d is V25() real ext-real Element of REAL
d mod 2 is V25() real ext-real integer set
N1 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
N0 . N1 is V25() real ext-real Element of REAL
f . N1 is V25() real ext-real Element of REAL
t * (f . N1) is V25() real ext-real Element of REAL
c is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() eventually-nonnegative Element of K19(K20(NAT,REAL))
Big_Oh c is functional non empty FUNCTION_DOMAIN of NAT , REAL
{ b₁ where b₁ is V1() V4( NAT ) V5( REAL ) Function-like V14( NAT ) quasi_total V35() V36() V37() Element of Funcs (NAT,REAL) : ex b₂ being V25() real ext-real Element of REAL ex b₃ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT st
( not b₂ <= 0 & ( for b₄ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT holds
( not b₃ <= b₄ or ( b₁ . b₄ <= b₂ * (c . b₄) & 0 <= b₁ . b₄ ) ) ) ) } is set
x is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() eventually-nonnegative Element of K19(K20(NAT,REAL))
Big_Oh x is functional non empty FUNCTION_DOMAIN of NAT , REAL
{ b₁ where b₁ is V1() V4( NAT ) V5( REAL ) Function-like V14( NAT ) quasi_total V35() V36() V37() Element of Funcs (NAT,REAL) : ex b₂ being V25() real ext-real Element of REAL ex b₃ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT st
( not b₂ <= 0 & ( for b₄ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT holds
( not b₃ <= b₄ or ( b₁ . b₄ <= b₂ * (x . b₄) & 0 <= b₁ . b₄ ) ) ) ) } is set
Big_Theta x is functional non empty FUNCTION_DOMAIN of NAT , REAL
Big_Omega x is functional non empty FUNCTION_DOMAIN of NAT , REAL
{ b₁ where b₁ is V1() V4( NAT ) V5( REAL ) Function-like V14( NAT ) quasi_total V35() V36() V37() Element of Funcs (NAT,REAL) : ex b₂ being V25() real ext-real Element of REAL ex b₃ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT st
( not b₂ <= 0 & ( for b₄ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT holds
( not b₃ <= b₄ or ( b₂ * (x . b₄) <= b₁ . b₄ & 0 <= b₁ . b₄ ) ) ) ) } is set
(Big_Oh x) /\ (Big_Omega x) is set
f is set
e is V1() V4( NAT ) V5( REAL ) Function-like V14( NAT ) quasi_total V35() V36() V37() Element of Funcs (NAT,REAL)
N0 is V25() real ext-real Element of REAL
t is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
N0 is V25() real ext-real Element of REAL
t is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
d is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
N1 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
e . N1 is V25() real ext-real Element of REAL
d is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
d is V25() real ext-real Element of REAL
N1 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
e is V1() V4( NAT ) V5( REAL ) Function-like V14( NAT ) quasi_total V35() V36() V37() Element of Funcs (NAT,REAL)
N0 is V25() real ext-real Element of REAL
t is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
N0 is V25() real ext-real Element of REAL
t is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
d is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
N1 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
e . N1 is V25() real ext-real Element of REAL
d is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
d is V25() real ext-real Element of REAL
N1 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
c is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() eventually-nonnegative Element of K19(K20(NAT,REAL))
x is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() eventually-nonnegative Element of K19(K20(NAT,REAL))
Big_Theta x is functional non empty FUNCTION_DOMAIN of NAT , REAL
Big_Oh x is functional non empty FUNCTION_DOMAIN of NAT , REAL
{ b₁ where b₁ is V1() V4( NAT ) V5( REAL ) Function-like V14( NAT ) quasi_total V35() V36() V37() Element of Funcs (NAT,REAL) : ex b₂ being V25() real ext-real Element of REAL ex b₃ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT st
( not b₂ <= 0 & ( for b₄ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT holds
( not b₃ <= b₄ or ( b₁ . b₄ <= b₂ * (x . b₄) & 0 <= b₁ . b₄ ) ) ) ) } is set
Big_Omega x is functional non empty FUNCTION_DOMAIN of NAT , REAL
{ b₁ where b₁ is V1() V4( NAT ) V5( REAL ) Function-like V14( NAT ) quasi_total V35() V36() V37() Element of Funcs (NAT,REAL) : ex b₂ being V25() real ext-real Element of REAL ex b₃ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT st
( not b₂ <= 0 & ( for b₄ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT holds
( not b₃ <= b₄ or ( b₂ * (x . b₄) <= b₁ . b₄ & 0 <= b₁ . b₄ ) ) ) ) } is set
(Big_Oh x) /\ (Big_Omega x) is set
Big_Theta c is functional non empty FUNCTION_DOMAIN of NAT , REAL
Big_Oh c is functional non empty FUNCTION_DOMAIN of NAT , REAL
{ b₁ where b₁ is V1() V4( NAT ) V5( REAL ) Function-like V14( NAT ) quasi_total V35() V36() V37() Element of Funcs (NAT,REAL) : ex b₂ being V25() real ext-real Element of REAL ex b₃ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT st
( not b₂ <= 0 & ( for b₄ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT holds
( not b₃ <= b₄ or ( b₁ . b₄ <= b₂ * (c . b₄) & 0 <= b₁ . b₄ ) ) ) ) } is set
Big_Omega c is functional non empty FUNCTION_DOMAIN of NAT , REAL
{ b₁ where b₁ is V1() V4( NAT ) V5( REAL ) Function-like V14( NAT ) quasi_total V35() V36() V37() Element of Funcs (NAT,REAL) : ex b₂ being V25() real ext-real Element of REAL ex b₃ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT st
( not b₂ <= 0 & ( for b₄ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT holds
( not b₃ <= b₄ or ( b₂ * (c . b₄) <= b₁ . b₄ & 0 <= b₁ . b₄ ) ) ) ) } is set
(Big_Oh c) /\ (Big_Omega c) is set
{ b₁ where b₁ is V1() V4( NAT ) V5( REAL ) Function-like V14( NAT ) quasi_total V35() V36() V37() Element of Funcs (NAT,REAL) : ex b₂, b₃ being V25() real ext-real Element of REAL ex b₄ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT st
( not b₂ <= 0 & not b₃ <= 0 & ( for b₅ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT holds
( not b₄ <= b₅ or ( b₃ * (x . b₅) <= b₁ . b₅ & b₁ . b₅ <= b₂ * (x . b₅) ) ) ) ) } is set
f is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
e is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
{ b₁ where b₁ is V1() V4( NAT ) V5( REAL ) Function-like V14( NAT ) quasi_total V35() V36() V37() Element of Funcs (NAT,REAL) : ex b₂, b₃ being V25() real ext-real Element of REAL ex b₄ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT st
( not b₂ <= 0 & not b₃ <= 0 & ( for b₅ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT holds
( not b₄ <= b₅ or ( b₃ * (c . b₅) <= b₁ . b₅ & b₁ . b₅ <= b₂ * (c . b₅) ) ) ) ) } is set
N0 is set
t is V1() V4( NAT ) V5( REAL ) Function-like V14( NAT ) quasi_total V35() V36() V37() Element of Funcs (NAT,REAL)
d is V25() real ext-real Element of REAL
N1 is V25() real ext-real Element of REAL
b is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
d is V25() real ext-real Element of REAL
N1 is V25() real ext-real Element of REAL
b is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
max (e,b) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
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
g9 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
c . g9 is V25() real ext-real Element of REAL
N1 * 0 is V25() real ext-real Element of REAL
N1 * (c . g9) is V25() real ext-real Element of REAL
t . g9 is V25() real ext-real Element of REAL
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
n is V25() real ext-real Element of REAL
g9 is V25() real ext-real Element of REAL
N0 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
t is V1() V4( NAT ) V5( REAL ) Function-like V14( NAT ) quasi_total V35() V36() V37() Element of Funcs (NAT,REAL)
d is V25() real ext-real Element of REAL
N1 is V25() real ext-real Element of REAL
b is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
d is V25() real ext-real Element of REAL
N1 is V25() real ext-real Element of REAL
b is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
max (f,b) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
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
g9 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
x . g9 is V25() real ext-real Element of REAL
N1 * 0 is V25() real ext-real Element of REAL
N1 * (x . g9) is V25() real ext-real Element of REAL
t . g9 is V25() real ext-real Element of REAL
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
n is V25() real ext-real Element of REAL
g9 is V25() real ext-real Element of REAL
N0 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
c is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
c ^2 is V25() real ext-real Element of REAL
c * c is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer set
(c ^2) - c is V25() real ext-real Element of REAL
- c is V25() real ext-real non positive integer set
(c ^2) + (- c) is V25() real ext-real set
((c ^2) - c) + 1 is V25() real ext-real Element of REAL
c + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(c + 1) ^2 is V25() real ext-real Element of REAL
(c + 1) * (c + 1) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer set
((c + 1) ^2) - (c + 1) is V25() real ext-real Element of REAL
- (c + 1) is V25() real ext-real non positive integer set
((c + 1) ^2) + (- (c + 1)) is V25() real ext-real set
(((c + 1) ^2) - (c + 1)) + 1 is V25() real ext-real Element of REAL
2 * c is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(((c ^2) - c) + 1) + (2 * c) is V25() real ext-real Element of REAL
0 + 0 is 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
0 ^2 is V25() real ext-real Element of REAL
0 * 0 is functional empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V25() real ext-real non positive non negative integer V49() V50() V51() V52() V53() V54() V55() FinSequence-membered set
(0 ^2) - 0 is V25() real ext-real Element of REAL
- 0 is functional empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V25() real ext-real non positive non negative integer V49() V50() V51() V52() V53() V54() V55() FinSequence-membered set
(0 ^2) + (- 0) is V25() real ext-real set
((0 ^2) - 0) + 1 is V25() real ext-real Element of REAL
c is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
c ^2 is V25() real ext-real Element of REAL
c * c is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer set
(c ^2) - c is V25() real ext-real Element of REAL
- c is V25() real ext-real non positive integer set
(c ^2) + (- c) is V25() real ext-real set
((c ^2) - c) + 1 is V25() real ext-real Element of REAL
c is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() Element of K19(K20(NAT,REAL))
lim c is V25() real ext-real Element of REAL
x is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() Element of K19(K20(NAT,REAL))
lim x is V25() real ext-real Element of REAL
f is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
e is V25() real ext-real Element of REAL
N0 is V25() real ext-real set
t is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
max (f,t) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
N1 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
b is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
c . b is V25() real ext-real Element of REAL
(c . b) - e is V25() real ext-real Element of REAL
- e is V25() real ext-real set
(c . b) + (- e) is V25() real ext-real set
abs ((c . b) - e) is V25() real ext-real Element of REAL
x . b is V25() real ext-real Element of REAL
(x . b) - e is V25() real ext-real Element of REAL
(x . b) + (- e) is V25() real ext-real set
abs ((x . b) - e) is V25() real ext-real Element of REAL
c is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
c ^2 is V25() real ext-real Element of REAL
c * c is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer set
(c ^2) - c is V25() real ext-real Element of REAL
- c is V25() real ext-real non positive integer set
(c ^2) + (- c) is V25() real ext-real set
((c ^2) - c) + 1 is V25() real ext-real Element of REAL
- (c ^2) is V25() real ext-real Element of REAL
- c is V25() real ext-real non positive integer Element of REAL
(- c) + 1 is V25() real ext-real integer Element of REAL
(c ^2) + ((- c) + 1) is V25() real ext-real Element of REAL
(- (c ^2)) + ((c ^2) + ((- c) + 1)) is V25() real ext-real Element of REAL
(c ^2) + (- (c ^2)) is V25() real ext-real Element of REAL
0 - (- c) is V25() real ext-real non negative integer Element of REAL
- (- c) is V25() real ext-real non negative integer set
0 + (- (- c)) is V25() real ext-real non negative integer set
c is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer set
c ^2 is V25() real ext-real set
c * c is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer set
(c ^2) - c is V25() real ext-real set
- c is V25() real ext-real non positive integer set
(c ^2) + (- c) is V25() real ext-real set
((c ^2) - c) + 1 is V25() real ext-real Element of REAL
2 * (((c ^2) - c) + 1) is V25() real ext-real Element of REAL
c + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(c + 1) ^2 is V25() real ext-real set
(c + 1) * (c + 1) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer set
((c + 1) ^2) - (c + 1) is V25() real ext-real set
- (c + 1) is V25() real ext-real non positive integer set
((c + 1) ^2) + (- (c + 1)) is V25() real ext-real set
(((c + 1) ^2) - (c + 1)) + 1 is V25() real ext-real Element of REAL
2 * ((((c + 1) ^2) - (c + 1)) + 1) is V25() real ext-real Element of REAL
2 * c is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(2 * c) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(c ^2) + ((2 * c) + 1) is V25() real ext-real Element of REAL
(2 * (((c ^2) - c) + 1)) + ((2 * c) + 1) is V25() real ext-real Element of REAL
2 * 1 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 + 2 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(2 * c) + 2 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 * (c ^2) is V25() real ext-real Element of REAL
(2 * (c ^2)) + 4 is V25() real ext-real Element of REAL
(2 * (c ^2)) + ((2 * c) + 2) is V25() real ext-real Element of REAL
(2 * (c ^2)) + 3 is V25() real ext-real Element of REAL
1 ^2 is V25() real ext-real set
1 * 1 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer set
(1 ^2) - 1 is V25() real ext-real set
- 1 is V25() real ext-real non positive integer set
(1 ^2) + (- 1) is V25() real ext-real set
((1 ^2) - 1) + 1 is V25() real ext-real Element of REAL
2 * (((1 ^2) - 1) + 1) is V25() real ext-real Element of REAL
c is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
c ^2 is V25() real ext-real Element of REAL
c * c is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer set
(c ^2) - c is V25() real ext-real Element of REAL
- c is V25() real ext-real non positive integer set
(c ^2) + (- c) is V25() real ext-real set
((c ^2) - c) + 1 is V25() real ext-real Element of REAL
2 * (((c ^2) - c) + 1) is V25() real ext-real Element of REAL
x is V25() real ext-real Element of REAL
1 + x is V25() real ext-real Element of REAL
log (2,(1 + x)) is V25() real ext-real Element of REAL
(x) is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() eventually-nonnegative eventually-positive eventually-nonzero Element of K19(K20(NAT,REAL))
() /" (x) is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() Element of K19(K20(NAT,REAL))
(x) " is V1() V4( NAT ) Function-like V14( NAT ) V35() V36() V37() set
() (#) ((x) ") is V1() V4( NAT ) Function-like V14( NAT ) V35() V36() V37() set
lim (() /" (x)) is V25() real ext-real Element of REAL
x + 1 is V25() real ext-real Element of REAL
0 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
log (2,(x + 1)) is V25() real ext-real Element of REAL
log (2,1) is V25() real ext-real Element of REAL
1 / 16 is V25() real ext-real non negative Element of REAL
16 " is non empty V25() real ext-real positive non negative set
1 * (16 ") is V25() real ext-real non negative set
(log (2,(1 + x))) * (1 / 16) is V25() real ext-real Element of REAL
(log (2,(1 + x))) * 0 is V25() real ext-real Element of REAL
(log (2,(1 + x))) / 16 is V25() real ext-real Element of REAL
(log (2,(1 + x))) * (16 ") is V25() real ext-real set
t is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
max (2,t) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
N1 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
b is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
b to_power x is V25() real ext-real Element of REAL
log (2,2) is V25() real ext-real Element of REAL
log (2,b) is V25() real ext-real Element of REAL
(() /" (x)) . b is V25() real ext-real Element of REAL
() . b is V25() real ext-real Element of REAL
(x) . b is V25() real ext-real Element of REAL
(() . b) / ((x) . b) is V25() real ext-real Element of REAL
((x) . b) " is V25() real ext-real set
(() . b) * (((x) . b) ") is V25() real ext-real set
(log (2,b)) / ((x) . b) is V25() real ext-real Element of REAL
(log (2,b)) * (((x) . b) ") is V25() real ext-real set
(log (2,b)) / (b to_power x) is V25() real ext-real Element of REAL
(b to_power x) " is V25() real ext-real set
(log (2,b)) * ((b to_power x) ") is V25() real ext-real set
b to_power 1 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
b |^ 1 is set
1 / (b to_power x) is V25() real ext-real Element of REAL
1 * ((b to_power x) ") is V25() real ext-real set
1 / (b to_power 1) is V25() real ext-real non negative Element of REAL
(b to_power 1) " is V25() real ext-real non negative set
1 * ((b to_power 1) ") is V25() real ext-real non negative set
1 / b is V25() real ext-real non negative Element of REAL
b " is V25() real ext-real non negative set
1 * (b ") is V25() real ext-real non negative set
(log (2,b)) * (1 / (b to_power x)) is V25() real ext-real Element of REAL
(log (2,b)) / b is V25() real ext-real Element of REAL
(log (2,b)) * (b ") is V25() real ext-real set
((() /" (x)) . b) - 0 is V25() real ext-real Element of REAL
- 0 is functional empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V25() real ext-real non positive non negative integer V49() V50() V51() V52() V53() V54() V55() FinSequence-membered set
((() /" (x)) . b) + (- 0) is V25() real ext-real set
abs (((() /" (x)) . b) - 0) is V25() real ext-real Element of REAL
b " is V25() real ext-real non negative Element of REAL
(log (2,b)) * (b ") is V25() real ext-real Element of REAL
0 * (b ") is functional empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V25() real ext-real non positive non negative integer V49() V50() V51() V52() V53() V54() V55() FinSequence-membered Element of REAL
1 / ((log (2,b)) / b) is V25() real ext-real Element of REAL
((log (2,b)) / b) " is V25() real ext-real set
1 * (((log (2,b)) / b) ") is V25() real ext-real set
1 / ((log (2,(1 + x))) / 16) is V25() real ext-real Element of REAL
((log (2,(1 + x))) / 16) " is V25() real ext-real set
1 * (((log (2,(1 + x))) / 16) ") is V25() real ext-real set
b / (log (2,b)) is V25() real ext-real Element of REAL
(log (2,b)) " is V25() real ext-real set
b * ((log (2,b)) ") is V25() real ext-real set
16 / (log (2,(1 + x))) is V25() real ext-real Element of REAL
(log (2,(1 + x))) " is V25() real ext-real set
16 * ((log (2,(1 + x))) ") is V25() real ext-real set
(log (2,(1 + x))) * (b / (log (2,b))) is V25() real ext-real Element of REAL
(16 / (log (2,(1 + x)))) * (log (2,(1 + x))) is V25() real ext-real Element of REAL
((log (2,(1 + x))) * (b / (log (2,b)))) * (log (2,b)) is V25() real ext-real Element of REAL
16 * (log (2,b)) is V25() real ext-real Element of REAL
(b / (log (2,b))) * (log (2,b)) is V25() real ext-real Element of REAL
(log (2,(1 + x))) * ((b / (log (2,b))) * (log (2,b))) is V25() real ext-real Element of REAL
(log (2,(1 + x))) * b is V25() real ext-real Element of REAL
8 + 8 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(8 + 8) * (log (2,b)) is V25() real ext-real Element of REAL
8 * (log (2,b)) is V25() real ext-real Element of REAL
((log (2,(1 + x))) * b) - (8 * (log (2,b))) is V25() real ext-real Element of REAL
- (8 * (log (2,b))) is V25() real ext-real set
((log (2,(1 + x))) * b) + (- (8 * (log (2,b)))) is V25() real ext-real set
(8 * (log (2,b))) + (8 * (log (2,b))) is V25() real ext-real Element of REAL
((8 * (log (2,b))) + (8 * (log (2,b)))) - (8 * (log (2,b))) is V25() real ext-real Element of REAL
((8 * (log (2,b))) + (8 * (log (2,b)))) + (- (8 * (log (2,b)))) is V25() real ext-real set
b * (log (2,(1 + x))) is V25() real ext-real Element of REAL
(b * (log (2,(1 + x)))) - (8 * (log (2,b))) is V25() real ext-real Element of REAL
(b * (log (2,(1 + x)))) + (- (8 * (log (2,b)))) is V25() real ext-real set
N1 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
d is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
x is V25() real ext-real Element of REAL
1 + x is V25() real ext-real Element of REAL
((1 + x)) is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() eventually-nonnegative eventually-positive eventually-nonzero Element of K19(K20(NAT,REAL))
Big_Oh ((1 + x)) is functional non empty FUNCTION_DOMAIN of NAT , REAL
{ b₁ where b₁ is V1() V4( NAT ) V5( REAL ) Function-like V14( NAT ) quasi_total V35() V36() V37() Element of Funcs (NAT,REAL) : ex b₂ being V25() real ext-real Element of REAL ex b₃ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT st
( not b₂ <= 0 & ( for b₄ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT holds
( not b₃ <= b₄ or ( b₁ . b₄ <= b₂ * (((1 + x)) . b₄) & 0 <= b₁ . b₄ ) ) ) ) } is set
f is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() Element of K19(K20(NAT,REAL))
(x) is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() eventually-nonnegative eventually-positive eventually-nonzero Element of K19(K20(NAT,REAL))
() /" (x) is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() Element of K19(K20(NAT,REAL))
(x) " is V1() V4( NAT ) Function-like V14( NAT ) V35() V36() V37() set
() (#) ((x) ") is V1() V4( NAT ) Function-like V14( NAT ) V35() V36() V37() set
lim (() /" (x)) is V25() real ext-real Element of REAL
t is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
f . t is V25() real ext-real Element of REAL
log (2,2) is V25() real ext-real Element of REAL
log (2,t) is V25() real ext-real Element of REAL
t * (log (2,t)) is V25() real ext-real Element of REAL
t * 0 is 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
t is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() eventually-nonnegative eventually-positive eventually-nonzero Element of K19(K20(NAT,REAL))
t /" ((1 + x)) is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() Element of K19(K20(NAT,REAL))
((1 + x)) " is V1() V4( NAT ) Function-like V14( NAT ) V35() V36() V37() set
t (#) (((1 + x)) ") is V1() V4( NAT ) Function-like V14( NAT ) V35() V36() V37() set
b is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(t /" ((1 + x))) . b is V25() real ext-real Element of REAL
(() /" (x)) . b is V25() real ext-real Element of REAL
t . b is V25() real ext-real Element of REAL
((1 + x)) . b is V25() real ext-real Element of REAL
(t . b) / (((1 + x)) . b) is V25() real ext-real Element of REAL
(((1 + x)) . b) " is V25() real ext-real set
(t . b) * ((((1 + x)) . b) ") is V25() real ext-real set
log (2,b) is V25() real ext-real Element of REAL
b * (log (2,b)) is V25() real ext-real Element of REAL
(b * (log (2,b))) / (((1 + x)) . b) is V25() real ext-real Element of REAL
(b * (log (2,b))) * ((((1 + x)) . b) ") is V25() real ext-real set
b to_power (1 + x) is V25() real ext-real Element of REAL
(b * (log (2,b))) / (b to_power (1 + x)) is V25() real ext-real Element of REAL
(b to_power (1 + x)) " is V25() real ext-real set
(b * (log (2,b))) * ((b to_power (1 + x)) ") is V25() real ext-real set
b to_power 1 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
b |^ 1 is set
(b to_power 1) * (log (2,b)) is V25() real ext-real Element of REAL
((b to_power 1) * (log (2,b))) / (b to_power (1 + x)) is V25() real ext-real Element of REAL
((b to_power 1) * (log (2,b))) * ((b to_power (1 + x)) ") is V25() real ext-real set
(b to_power (1 + x)) " is V25() real ext-real Element of REAL
((b to_power 1) * (log (2,b))) * ((b to_power (1 + x)) ") is V25() real ext-real Element of REAL
(b to_power 1) * ((b to_power (1 + x)) ") is V25() real ext-real Element of REAL
(log (2,b)) * ((b to_power 1) * ((b to_power (1 + x)) ")) is V25() real ext-real Element of REAL
(b to_power 1) / (b to_power (1 + x)) is V25() real ext-real Element of REAL
(b to_power 1) * ((b to_power (1 + x)) ") is V25() real ext-real set
(log (2,b)) * ((b to_power 1) / (b to_power (1 + x))) is V25() real ext-real Element of REAL
1 - (1 + x) is V25() real ext-real Element of REAL
- (1 + x) is V25() real ext-real set
1 + (- (1 + x)) is V25() real ext-real set
b to_power (1 - (1 + x)) is V25() real ext-real Element of REAL
(log (2,b)) * (b to_power (1 - (1 + x))) is V25() real ext-real Element of REAL
- x is V25() real ext-real Element of REAL
(- 1) + (- x) is V25() real ext-real Element of REAL
1 + ((- 1) + (- x)) is V25() real ext-real Element of REAL
b to_power (1 + ((- 1) + (- x))) is V25() real ext-real Element of REAL
(log (2,b)) * (b to_power (1 + ((- 1) + (- x)))) is V25() real ext-real Element of REAL
b to_power x is V25() real ext-real Element of REAL
1 / (b to_power x) is V25() real ext-real Element of REAL
(b to_power x) " is V25() real ext-real set
1 * ((b to_power x) ") is V25() real ext-real set
(log (2,b)) * (1 / (b to_power x)) is V25() real ext-real Element of REAL
(log (2,b)) / (b to_power x) is V25() real ext-real Element of REAL
(log (2,b)) * ((b to_power x) ") is V25() real ext-real set
() . b is V25() real ext-real Element of REAL
(() . b) / (b to_power x) is V25() real ext-real Element of REAL
(() . b) * ((b to_power x) ") is V25() real ext-real set
(x) . b is V25() real ext-real Element of REAL
(() . b) / ((x) . b) is V25() real ext-real Element of REAL
((x) . b) " is V25() real ext-real set
(() . b) * (((x) . b) ") is V25() real ext-real set
lim (t /" ((1 + x))) is V25() real ext-real Element of REAL
Big_Oh t is functional non empty FUNCTION_DOMAIN of NAT , REAL
{ b₁ where b₁ is V1() V4( NAT ) V5( REAL ) Function-like V14( NAT ) quasi_total V35() V36() V37() Element of Funcs (NAT,REAL) : ex b₂ being V25() real ext-real Element of REAL ex b₃ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT st
( not b₂ <= 0 & ( for b₄ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT holds
( not b₃ <= b₄ or ( b₁ . b₄ <= b₂ * (t . b₄) & 0 <= b₁ . b₄ ) ) ) ) } is set
Big_Omega ((1 + x)) is functional non empty FUNCTION_DOMAIN of NAT , REAL
{ b₁ where b₁ is V1() V4( NAT ) V5( REAL ) Function-like V14( NAT ) quasi_total V35() V36() V37() Element of Funcs (NAT,REAL) : ex b₂ being V25() real ext-real Element of REAL ex b₃ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT st
( not b₂ <= 0 & ( for b₄ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT holds
( not b₃ <= b₄ or ( b₂ * (((1 + x)) . b₄) <= b₁ . b₄ & 0 <= b₁ . b₄ ) ) ) ) } is set
x is V25() real ext-real Element of REAL
1 + x is V25() real ext-real Element of REAL
((1 + x)) is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() eventually-nonnegative eventually-positive eventually-nonzero Element of K19(K20(NAT,REAL))
Big_Oh ((1 + x)) is functional non empty FUNCTION_DOMAIN of NAT , REAL
{ b₁ where b₁ is V1() V4( NAT ) V5( REAL ) Function-like V14( NAT ) quasi_total V35() V36() V37() Element of Funcs (NAT,REAL) : ex b₂ being V25() real ext-real Element of REAL ex b₃ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT st
( not b₂ <= 0 & ( for b₄ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT holds
( not b₃ <= b₄ or ( b₁ . b₄ <= b₂ * (((1 + x)) . b₄) & 0 <= b₁ . b₄ ) ) ) ) } is set
f is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() Element of K19(K20(NAT,REAL))
1 - x is V25() real ext-real Element of REAL
- x is V25() real ext-real set
1 + (- x) is V25() real ext-real set
((1 - x)) is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() eventually-nonnegative eventually-positive eventually-nonzero Element of K19(K20(NAT,REAL))
() /" ((1 - x)) is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() Element of K19(K20(NAT,REAL))
((1 - x)) " is V1() V4( NAT ) Function-like V14( NAT ) V35() V36() V37() set
() (#) (((1 - x)) ") is V1() V4( NAT ) Function-like V14( NAT ) V35() V36() V37() set
((1 + x)) /" f is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() Element of K19(K20(NAT,REAL))
f " is V1() V4( NAT ) Function-like V14( NAT ) V35() V36() V37() set
((1 + x)) (#) (f ") is V1() V4( NAT ) Function-like V14( NAT ) V35() V36() V37() set
N1 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
f . N1 is V25() real ext-real Element of REAL
log (2,2) is V25() real ext-real Element of REAL
log (2,N1) is V25() real ext-real Element of REAL
N1 to_power 2 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
N1 |^ 2 is set
(N1 to_power 2) / (log (2,N1)) is V25() real ext-real Element of REAL
(log (2,N1)) " is V25() real ext-real set
(N1 to_power 2) * ((log (2,N1)) ") is V25() real ext-real set
(log (2,N1)) " is V25() real ext-real Element of REAL
(N1 to_power 2) * ((log (2,N1)) ") is V25() real ext-real Element of REAL
(N1 to_power 2) * 0 is 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
(1 + x) - 2 is V25() real ext-real Element of REAL
- 2 is V25() real ext-real non positive integer set
(1 + x) + (- 2) is V25() real ext-real set
x - 1 is V25() real ext-real Element of REAL
- 1 is V25() real ext-real non positive integer set
x + (- 1) is V25() real ext-real set
b is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(((1 + x)) /" f) . b is V25() real ext-real Element of REAL
(() /" ((1 - x))) . b is V25() real ext-real Element of REAL
((1 + x)) . b is V25() real ext-real Element of REAL
N1 is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() eventually-nonnegative eventually-positive eventually-nonzero Element of K19(K20(NAT,REAL))
N1 . b is V25() real ext-real Element of REAL
(((1 + x)) . b) / (N1 . b) is V25() real ext-real Element of REAL
(N1 . b) " is V25() real ext-real set
(((1 + x)) . b) * ((N1 . b) ") is V25() real ext-real set
b to_power (1 + x) is V25() real ext-real Element of REAL
(b to_power (1 + x)) / (N1 . b) is V25() real ext-real Element of REAL
(b to_power (1 + x)) * ((N1 . b) ") is V25() real ext-real set
b to_power 2 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
b |^ 2 is set
log (2,b) is V25() real ext-real Element of REAL
(b to_power 2) / (log (2,b)) is V25() real ext-real Element of REAL
(log (2,b)) " is V25() real ext-real set
(b to_power 2) * ((log (2,b)) ") is V25() real ext-real set
(b to_power (1 + x)) / ((b to_power 2) / (log (2,b))) is V25() real ext-real Element of REAL
((b to_power 2) / (log (2,b))) " is V25() real ext-real set
(b to_power (1 + x)) * (((b to_power 2) / (log (2,b))) ") is V25() real ext-real set
((b to_power 2) / (log (2,b))) " is V25() real ext-real Element of REAL
(b to_power (1 + x)) * (((b to_power 2) / (log (2,b))) ") is V25() real ext-real Element of REAL
(log (2,b)) / (b to_power 2) is V25() real ext-real Element of REAL
(b to_power 2) " is V25() real ext-real non negative set
(log (2,b)) * ((b to_power 2) ") is V25() real ext-real set
(b to_power (1 + x)) * ((log (2,b)) / (b to_power 2)) is V25() real ext-real Element of REAL
(b to_power 2) " is V25() real ext-real non negative Element of REAL
(log (2,b)) * ((b to_power 2) ") is V25() real ext-real Element of REAL
(b to_power (1 + x)) * ((log (2,b)) * ((b to_power 2) ")) is V25() real ext-real Element of REAL
(b to_power (1 + x)) * ((b to_power 2) ") is V25() real ext-real Element of REAL
((b to_power (1 + x)) * ((b to_power 2) ")) * (log (2,b)) is V25() real ext-real Element of REAL
(b to_power (1 + x)) / (b to_power 2) is V25() real ext-real Element of REAL
(b to_power (1 + x)) * ((b to_power 2) ") is V25() real ext-real set
((b to_power (1 + x)) / (b to_power 2)) * (log (2,b)) is V25() real ext-real Element of REAL
- (1 - x) is V25() real ext-real Element of REAL
b to_power (- (1 - x)) is V25() real ext-real Element of REAL
(b to_power (- (1 - x))) * (log (2,b)) is V25() real ext-real Element of REAL
b to_power (1 - x) is V25() real ext-real Element of REAL
1 / (b to_power (1 - x)) is V25() real ext-real Element of REAL
(b to_power (1 - x)) " is V25() real ext-real set
1 * ((b to_power (1 - x)) ") is V25() real ext-real set
(log (2,b)) * (1 / (b to_power (1 - x))) is V25() real ext-real Element of REAL
(log (2,b)) / (b to_power (1 - x)) is V25() real ext-real Element of REAL
(log (2,b)) * ((b to_power (1 - x)) ") is V25() real ext-real set
() . b is V25() real ext-real Element of REAL
(() . b) / (b to_power (1 - x)) is V25() real ext-real Element of REAL
(() . b) * ((b to_power (1 - x)) ") is V25() real ext-real set
((1 - x)) . b is V25() real ext-real Element of REAL
(() . b) / (((1 - x)) . b) is V25() real ext-real Element of REAL
(((1 - x)) . b) " is V25() real ext-real set
(() . b) * ((((1 - x)) . b) ") is V25() real ext-real set
N1 is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() eventually-nonnegative eventually-positive eventually-nonzero Element of K19(K20(NAT,REAL))
Big_Oh N1 is functional non empty FUNCTION_DOMAIN of NAT , REAL
{ b₁ where b₁ is V1() V4( NAT ) V5( REAL ) Function-like V14( NAT ) quasi_total V35() V36() V37() Element of Funcs (NAT,REAL) : ex b₂ being V25() real ext-real Element of REAL ex b₃ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT st
( not b₂ <= 0 & ( for b₄ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT holds
( not b₃ <= b₄ or ( b₁ . b₄ <= b₂ * (N1 . b₄) & 0 <= b₁ . b₄ ) ) ) ) } is set
0 + x is V25() real ext-real Element of REAL
lim (() /" ((1 - x))) is V25() real ext-real Element of REAL
lim (((1 + x)) /" f) is V25() real ext-real Element of REAL
Big_Omega N1 is functional non empty FUNCTION_DOMAIN of NAT , REAL
{ b₁ where b₁ is V1() V4( NAT ) V5( REAL ) Function-like V14( NAT ) quasi_total V35() V36() V37() Element of Funcs (NAT,REAL) : ex b₂ being V25() real ext-real Element of REAL ex b₃ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT st
( not b₂ <= 0 & ( for b₄ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT holds
( not b₃ <= b₄ or ( b₂ * (N1 . b₄) <= b₁ . b₄ & 0 <= b₁ . b₄ ) ) ) ) } is set
(8) is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() eventually-nonnegative eventually-positive eventually-nonzero Element of K19(K20(NAT,REAL))
Big_Oh (8) is functional non empty FUNCTION_DOMAIN of NAT , REAL
{ b₁ where b₁ is V1() V4( NAT ) V5( REAL ) Function-like V14( NAT ) quasi_total V35() V36() V37() Element of Funcs (NAT,REAL) : ex b₂ being V25() real ext-real Element of REAL ex b₃ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT st
( not b₂ <= 0 & ( for b₄ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT holds
( not b₃ <= b₄ or ( b₁ . b₄ <= b₂ * ((8) . b₄) & 0 <= b₁ . b₄ ) ) ) ) } is set
x is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() Element of K19(K20(NAT,REAL))
f is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
x . f is V25() real ext-real Element of REAL
log (2,2) is V25() real ext-real Element of REAL
log (2,f) is V25() real ext-real Element of REAL
f to_power 2 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
f |^ 2 is set
(f to_power 2) / (log (2,f)) is V25() real ext-real Element of REAL
(log (2,f)) " is V25() real ext-real set
(f to_power 2) * ((log (2,f)) ") is V25() real ext-real set
(log (2,f)) " is V25() real ext-real Element of REAL
(f to_power 2) * ((log (2,f)) ") is V25() real ext-real Element of REAL
(f to_power 2) * 0 is 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
x /" (8) is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() Element of K19(K20(NAT,REAL))
(8) " is V1() V4( NAT ) Function-like V14( NAT ) V35() V36() V37() set
x (#) ((8) ") is V1() V4( NAT ) Function-like V14( NAT ) V35() V36() V37() set
log (2,2) is V25() real ext-real Element of REAL
N0 is V25() real ext-real set
1 / 6 is V25() real ext-real non negative Element of REAL
6 " is non empty V25() real ext-real positive non negative set
1 * (6 ") is V25() real ext-real non negative set
- (1 / 6) is V25() real ext-real non positive Element of REAL
N0 to_power (- (1 / 6)) is V25() real ext-real set
[/(N0 to_power (- (1 / 6)))\] is V25() real ext-real integer set
t is V25() real ext-real Element of REAL
t to_power (- (1 / 6)) is V25() real ext-real Element of REAL
[/(t to_power (- (1 / 6)))\] is V25() real ext-real integer set
max (3,[/(t to_power (- (1 / 6)))\]) is V25() real ext-real set
N1 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
b is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
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
(x /" (8)) . N is V25() real ext-real Element of REAL
e is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() eventually-nonnegative eventually-positive eventually-nonzero Element of K19(K20(NAT,REAL))
e . N is V25() real ext-real Element of REAL
(8) . N is V25() real ext-real Element of REAL
(e . N) / ((8) . N) is V25() real ext-real Element of REAL
((8) . N) " is V25() real ext-real set
(e . N) * (((8) . N) ") is V25() real ext-real set
N to_power 2 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
N |^ 2 is set
log (2,N) is V25() real ext-real Element of REAL
(N to_power 2) / (log (2,N)) is V25() real ext-real Element of REAL
(log (2,N)) " is V25() real ext-real set
(N to_power 2) * ((log (2,N)) ") is V25() real ext-real set
((N to_power 2) / (log (2,N))) / ((8) . N) is V25() real ext-real Element of REAL
((N to_power 2) / (log (2,N))) * (((8) . N) ") is V25() real ext-real set
N to_power 8 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
N |^ 8 is set
((N to_power 2) / (log (2,N))) / (N to_power 8) is V25() real ext-real Element of REAL
(N to_power 8) " is V25() real ext-real non negative set
((N to_power 2) / (log (2,N))) * ((N to_power 8) ") is V25() real ext-real set
(log (2,N)) " is V25() real ext-real Element of REAL
(N to_power 2) * ((log (2,N)) ") is V25() real ext-real Element of REAL
((N to_power 2) * ((log (2,N)) ")) / (N to_power 8) is V25() real ext-real Element of REAL
((N to_power 2) * ((log (2,N)) ")) * ((N to_power 8) ") is V25() real ext-real set
((log (2,N)) ") * (N to_power 2) is V25() real ext-real Element of REAL
(N to_power 8) " is V25() real ext-real non negative Element of REAL
(((log (2,N)) ") * (N to_power 2)) * ((N to_power 8) ") is V25() real ext-real Element of REAL
(N to_power 2) * ((N to_power 8) ") is V25() real ext-real non negative Element of REAL
((log (2,N)) ") * ((N to_power 2) * ((N to_power 8) ")) is V25() real ext-real Element of REAL
(N to_power 2) / (N to_power 8) is V25() real ext-real non negative Element of REAL
(N to_power 2) * ((N to_power 8) ") is V25() real ext-real non negative set
((log (2,N)) ") * ((N to_power 2) / (N to_power 8)) is V25() real ext-real Element of REAL
2 - 8 is V25() real ext-real integer Element of REAL
- 8 is V25() real ext-real non positive integer set
2 + (- 8) is V25() real ext-real integer set
N to_power (2 - 8) is V25() real ext-real Element of REAL
((log (2,N)) ") * (N to_power (2 - 8)) is V25() real ext-real Element of REAL
- 6 is V25() real ext-real non positive integer Element of REAL
N to_power (- 6) is V25() real ext-real Element of REAL
((log (2,N)) ") * (N to_power (- 6)) is V25() real ext-real Element of REAL
N to_power 6 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
N |^ 6 is set
1 / (N to_power 6) is V25() real ext-real non negative Element of REAL
(N to_power 6) " is V25() real ext-real non negative set
1 * ((N to_power 6) ") is V25() real ext-real non negative set
((log (2,N)) ") * (1 / (N to_power 6)) is V25() real ext-real Element of REAL
1 / (log (2,N)) is V25() real ext-real Element of REAL
1 * ((log (2,N)) ") is V25() real ext-real set
(1 / (N to_power 6)) * (1 / (log (2,N))) is V25() real ext-real Element of REAL
(N to_power 6) * (log (2,N)) is V25() real ext-real Element of REAL
1 / ((N to_power 6) * (log (2,N))) is V25() real ext-real Element of REAL
((N to_power 6) * (log (2,N))) " is V25() real ext-real set
1 * (((N to_power 6) * (log (2,N))) ") is V25() real ext-real set
(N0 to_power (- (1 / 6))) to_power 6 is V25() real ext-real set
(N0 to_power (- (1 / 6))) |^ 6 is set
(- (1 / 6)) * 6 is V25() real ext-real non positive Element of REAL
t to_power ((- (1 / 6)) * 6) is V25() real ext-real Element of REAL
t to_power (- 1) is V25() real ext-real Element of REAL
N0 to_power (- 1) is V25() real ext-real set
1 / (N0 to_power (- 1)) is V25() real ext-real Element of REAL
(N0 to_power (- 1)) " is V25() real ext-real set
1 * ((N0 to_power (- 1)) ") is V25() real ext-real set
t to_power 1 is V25() real ext-real Element of REAL
t |^ 1 is set
1 / (t to_power 1) is V25() real ext-real Element of REAL
(t to_power 1) " is V25() real ext-real set
1 * ((t to_power 1) ") is V25() real ext-real set
1 / (1 / (t to_power 1)) is V25() real ext-real Element of REAL
(1 / (t to_power 1)) " is V25() real ext-real set
1 * ((1 / (t to_power 1)) ") is V25() real ext-real set
(N to_power 6) * 1 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
((x /" (8)) . N) - 0 is V25() real ext-real Element of REAL
- 0 is functional empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V25() real ext-real non positive non negative integer V49() V50() V51() V52() V53() V54() V55() FinSequence-membered set
((x /" (8)) . N) + (- 0) is V25() real ext-real set
abs (((x /" (8)) . N) - 0) is V25() real ext-real Element of REAL
lim (x /" (8)) is V25() real ext-real Element of REAL
Big_Oh e is functional non empty FUNCTION_DOMAIN of NAT , REAL
{ b₁ where b₁ is V1() V4( NAT ) V5( REAL ) Function-like V14( NAT ) quasi_total V35() V36() V37() Element of Funcs (NAT,REAL) : ex b₂ being V25() real ext-real Element of REAL ex b₃ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT st
( not b₂ <= 0 & ( for b₄ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT holds
( not b₃ <= b₄ or ( b₁ . b₄ <= b₂ * (e . b₄) & 0 <= b₁ . b₄ ) ) ) ) } is set
Big_Omega (8) is functional non empty FUNCTION_DOMAIN of NAT , REAL
{ b₁ where b₁ is V1() V4( NAT ) V5( REAL ) Function-like V14( NAT ) quasi_total V35() V36() V37() Element of Funcs (NAT,REAL) : ex b₂ being V25() real ext-real Element of REAL ex b₃ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT st
( not b₂ <= 0 & ( for b₄ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT holds
( not b₃ <= b₄ or ( b₂ * ((8) . b₄) <= b₁ . b₄ & 0 <= b₁ . b₄ ) ) ) ) } is set
c is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() Element of K19(K20(NAT,REAL))
x is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
c . x is V25() real ext-real Element of REAL
x ^2 is V25() real ext-real Element of REAL
x * x is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer set
(x ^2) - x is V25() real ext-real Element of REAL
- x is V25() real ext-real non positive integer set
(x ^2) + (- x) is V25() real ext-real set
((x ^2) - x) + 1 is V25() real ext-real Element of REAL
(((x ^2) - x) + 1) to_power 4 is V25() real ext-real Element of REAL
(((x ^2) - x) + 1) |^ 4 is set
x is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() eventually-nonnegative eventually-positive eventually-nonzero Element of K19(K20(NAT,REAL))
Big_Oh x is functional non empty FUNCTION_DOMAIN of NAT , REAL
{ b₁ where b₁ is V1() V4( NAT ) V5( REAL ) Function-like V14( NAT ) quasi_total V35() V36() V37() Element of Funcs (NAT,REAL) : ex b₂ being V25() real ext-real Element of REAL ex b₃ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT st
( not b₂ <= 0 & ( for b₄ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT holds
( not b₃ <= b₄ or ( b₁ . b₄ <= b₂ * (x . b₄) & 0 <= b₁ . b₄ ) ) ) ) } is set
e is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
x . e is V25() real ext-real Element of REAL
e ^2 is V25() real ext-real Element of REAL
e * e is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer set
(e ^2) - e is V25() real ext-real Element of REAL
- e is V25() real ext-real non positive integer set
(e ^2) + (- e) is V25() real ext-real set
((e ^2) - e) + 1 is V25() real ext-real Element of REAL
(((e ^2) - e) + 1) to_power 4 is V25() real ext-real Element of REAL
(((e ^2) - e) + 1) |^ 4 is set
(8) . e is V25() real ext-real Element of REAL
2 * 4 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
e to_power (2 * 4) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
e |^ (2 * 4) is set
e to_power 2 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
e |^ 2 is set
(e to_power 2) to_power 4 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(e to_power 2) |^ 4 is set
(e ^2) to_power 4 is V25() real ext-real Element of REAL
(e ^2) |^ 4 is set
1 * ((8) . e) is V25() real ext-real Element of REAL
e is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
x . e is V25() real ext-real Element of REAL
e ^2 is V25() real ext-real Element of REAL
e * e is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer set
(e ^2) - e is V25() real ext-real Element of REAL
- e is V25() real ext-real non positive integer set
(e ^2) + (- e) is V25() real ext-real set
((e ^2) - e) + 1 is V25() real ext-real Element of REAL
(((e ^2) - e) + 1) to_power 4 is V25() real ext-real Element of REAL
(((e ^2) - e) + 1) |^ 4 is set
(8) . e is V25() real ext-real Element of REAL
e to_power (2 * 4) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
e |^ (2 * 4) is set
e to_power 2 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
e |^ 2 is set
(e to_power 2) to_power 4 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(e to_power 2) |^ 4 is set
(e ^2) to_power 4 is V25() real ext-real Element of REAL
(e ^2) |^ 4 is set
e * e is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
e * 0 is 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
2 * (((e ^2) - e) + 1) is V25() real ext-real Element of REAL
(2 * (((e ^2) - e) + 1)) to_power 4 is V25() real ext-real Element of REAL
(2 * (((e ^2) - e) + 1)) |^ 4 is set
16 * (x . e) is V25() real ext-real Element of REAL
x is V25() real ext-real Element of REAL
1 + x is V25() real ext-real Element of REAL
((1 + x),1,0) is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() Element of K19(K20(NAT,REAL))
log (2,(1 + x)) is V25() real ext-real Element of REAL
f is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(8) /" ((1 + x),1,0) is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() Element of K19(K20(NAT,REAL))
((1 + x),1,0) " is V1() V4( NAT ) Function-like V14( NAT ) V35() V36() V37() set
(8) (#) (((1 + x),1,0) ") is V1() V4( NAT ) Function-like V14( NAT ) V35() V36() V37() set
t is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() eventually-nonnegative eventually-positive eventually-nonzero Element of K19(K20(NAT,REAL))
Big_Oh t is functional non empty FUNCTION_DOMAIN of NAT , REAL
{ b₁ where b₁ is V1() V4( NAT ) V5( REAL ) Function-like V14( NAT ) quasi_total V35() V36() V37() Element of Funcs (NAT,REAL) : ex b₂ being V25() real ext-real Element of REAL ex b₃ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT st
( not b₂ <= 0 & ( for b₄ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT holds
( not b₃ <= b₄ or ( b₁ . b₄ <= b₂ * (t . b₄) & 0 <= b₁ . b₄ ) ) ) ) } is set
d is V25() real ext-real set
N1 is V25() real ext-real Element of REAL
1 / N1 is V25() real ext-real Element of REAL
N1 " is V25() real ext-real set
1 * (N1 ") is V25() real ext-real set
1 / 8 is V25() real ext-real non negative Element of REAL
8 " is non empty V25() real ext-real positive non negative set
1 * (8 ") is V25() real ext-real non negative set
(1 / N1) to_power (1 / 8) is V25() real ext-real Element of REAL
[/((1 / N1) to_power (1 / 8))\] is V25() real ext-real integer set
max ([/((1 / N1) to_power (1 / 8))\],2) is V25() real ext-real set
max (f,(max ([/((1 / N1) to_power (1 / 8))\],2))) is V25() real ext-real set
1 / d is V25() real ext-real Element of REAL
d " is V25() real ext-real set
1 * (d ") is V25() real ext-real set
(1 / d) to_power (1 / 8) is V25() real ext-real Element of REAL
[/((1 / d) to_power (1 / 8))\] is V25() real ext-real integer set
max ([/((1 / d) to_power (1 / 8))\],2) is V25() real ext-real set
1 / d is V25() real ext-real Element of REAL
d " is V25() real ext-real set
1 * (d ") is V25() real ext-real set
(1 / d) to_power (1 / 8) is V25() real ext-real Element of REAL
[/((1 / d) to_power (1 / 8))\] is V25() real ext-real integer set
max ([/((1 / d) to_power (1 / 8))\],2) is V25() real ext-real set
1 / d is V25() real ext-real Element of REAL
d " is V25() real ext-real set
1 * (d ") is V25() real ext-real set
(1 / d) to_power (1 / 8) is V25() real ext-real Element of REAL
[/((1 / d) to_power (1 / 8))\] is V25() real ext-real integer set
max ([/((1 / d) to_power (1 / 8))\],2) is V25() real ext-real set
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
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
g9 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
g9 * (log (2,(1 + x))) is V25() real ext-real Element of REAL
log (2,g9) is V25() real ext-real Element of REAL
8 * (log (2,g9)) is V25() real ext-real Element of REAL
(g9 * (log (2,(1 + x)))) - (8 * (log (2,g9))) is V25() real ext-real Element of REAL
- (8 * (log (2,g9))) is V25() real ext-real set
(g9 * (log (2,(1 + x)))) + (- (8 * (log (2,g9)))) is V25() real ext-real set
2 to_power ((g9 * (log (2,(1 + x)))) - (8 * (log (2,g9)))) is V25() real ext-real Element of REAL
2 to_power (8 * (log (2,g9))) is V25() real ext-real Element of REAL
t . g9 is V25() real ext-real Element of REAL
1 * g9 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(1 * g9) + 0 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(1 + x) to_power ((1 * g9) + 0) is V25() real ext-real Element of REAL
(1 + x) |^ ((1 * g9) + 0) is set
((8) /" ((1 + x),1,0)) . g9 is V25() real ext-real Element of REAL
(8) . g9 is V25() real ext-real Element of REAL
((8) . g9) / (t . g9) is V25() real ext-real Element of REAL
(t . g9) " is V25() real ext-real set
((8) . g9) * ((t . g9) ") is V25() real ext-real set
g9 to_power 8 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
g9 |^ 8 is set
(1 + x) to_power g9 is V25() real ext-real Element of REAL
(1 + x) |^ g9 is set
(g9 to_power 8) / ((1 + x) to_power g9) is V25() real ext-real Element of REAL
((1 + x) to_power g9) " is V25() real ext-real set
(g9 to_power 8) * (((1 + x) to_power g9) ") is V25() real ext-real set
(2 to_power (8 * (log (2,g9)))) / ((1 + x) to_power g9) is V25() real ext-real Element of REAL
(2 to_power (8 * (log (2,g9)))) * (((1 + x) to_power g9) ") is V25() real ext-real set
2 to_power (g9 * (log (2,(1 + x)))) is V25() real ext-real Element of REAL
(2 to_power (8 * (log (2,g9)))) / (2 to_power (g9 * (log (2,(1 + x))))) is V25() real ext-real Element of REAL
(2 to_power (g9 * (log (2,(1 + x))))) " is V25() real ext-real set
(2 to_power (8 * (log (2,g9)))) * ((2 to_power (g9 * (log (2,(1 + x))))) ") is V25() real ext-real set
(8 * (log (2,g9))) - (g9 * (log (2,(1 + x)))) is V25() real ext-real Element of REAL
- (g9 * (log (2,(1 + x)))) is V25() real ext-real set
(8 * (log (2,g9))) + (- (g9 * (log (2,(1 + x))))) is V25() real ext-real set
2 to_power ((8 * (log (2,g9))) - (g9 * (log (2,(1 + x))))) is V25() real ext-real Element of REAL
- ((g9 * (log (2,(1 + x)))) - (8 * (log (2,g9)))) is V25() real ext-real Element of REAL
2 to_power (- ((g9 * (log (2,(1 + x)))) - (8 * (log (2,g9))))) is V25() real ext-real Element of REAL
((1 / d) to_power (1 / 8)) to_power 8 is V25() real ext-real Element of REAL
((1 / d) to_power (1 / 8)) |^ 8 is set
(1 / 8) * 8 is V25() real ext-real non negative Element of REAL
(1 / N1) to_power ((1 / 8) * 8) is V25() real ext-real Element of REAL
1 / (g9 to_power 8) is V25() real ext-real non negative Element of REAL
(g9 to_power 8) " is V25() real ext-real non negative set
1 * ((g9 to_power 8) ") is V25() real ext-real non negative set
1 / (d ") is V25() real ext-real Element of REAL
(d ") " is V25() real ext-real set
1 * ((d ") ") is V25() real ext-real set
1 / (2 to_power (8 * (log (2,g9)))) is V25() real ext-real Element of REAL
(2 to_power (8 * (log (2,g9)))) " is V25() real ext-real set
1 * ((2 to_power (8 * (log (2,g9)))) ") is V25() real ext-real set
- (8 * (log (2,g9))) is V25() real ext-real Element of REAL
2 to_power (- (8 * (log (2,g9)))) is V25() real ext-real Element of REAL
1 / (2 to_power ((g9 * (log (2,(1 + x)))) - (8 * (log (2,g9))))) is V25() real ext-real Element of REAL
(2 to_power ((g9 * (log (2,(1 + x)))) - (8 * (log (2,g9))))) " is V25() real ext-real set
1 * ((2 to_power ((g9 * (log (2,(1 + x)))) - (8 * (log (2,g9))))) ") is V25() real ext-real set
(((8) /" ((1 + x),1,0)) . g9) - 0 is V25() real ext-real Element of REAL
- 0 is functional empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V25() real ext-real non positive non negative integer V49() V50() V51() V52() V53() V54() V55() FinSequence-membered set
(((8) /" ((1 + x),1,0)) . g9) + (- 0) is V25() real ext-real set
abs ((((8) /" ((1 + x),1,0)) . g9) - 0) is V25() real ext-real Element of REAL
lim ((8) /" ((1 + x),1,0)) is V25() real ext-real Element of REAL
Big_Omega t is functional non empty FUNCTION_DOMAIN of NAT , REAL
{ b₁ where b₁ is V1() V4( NAT ) V5( REAL ) Function-like V14( NAT ) quasi_total V35() V36() V37() Element of Funcs (NAT,REAL) : ex b₂ being V25() real ext-real Element of REAL ex b₃ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT st
( not b₂ <= 0 & ( for b₄ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT holds
( not b₃ <= b₄ or ( b₂ * (t . b₄) <= b₁ . b₄ & 0 <= b₁ . b₄ ) ) ) ) } is set
2 to_power 12 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 |^ 12 is set
4096 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
6 + 6 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 to_power (6 + 6) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 |^ (6 + 6) is set
64 * 64 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
c is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer set
c ^2 is V25() real ext-real set
c * c is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer set
2 * c is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(2 * c) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
c + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(c + 1) ^2 is V25() real ext-real set
(c + 1) * (c + 1) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer set
2 * (c + 1) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(2 * (c + 1)) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
c + c is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer set
1 + 0 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(2 * c) + (2 * (c + 1)) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
1 + (2 * (c + 1)) is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
c + (c + 1) is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(c ^2) + (c + (c + 1)) is V25() real ext-real Element of REAL
((2 * c) + 1) + (c + (c + 1)) is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(c + 1) ^2 is V25() real ext-real Element of REAL
3 ^2 is V25() real ext-real set
3 * 3 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer set
2 * 3 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(2 * 3) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
c is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer set
c ^2 is V25() real ext-real set
c * c is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer set
2 * c is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(2 * c) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
c is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer set
c - 1 is V25() real ext-real integer Element of REAL
- 1 is V25() real ext-real non positive integer set
c + (- 1) is V25() real ext-real integer set
2 to_power (c - 1) is V25() real ext-real Element of REAL
2 * c is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(2 * c) ^2 is V25() real ext-real Element of REAL
(2 * c) * (2 * c) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer set
c + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(c + 1) - 1 is V25() real ext-real integer Element of REAL
(c + 1) + (- 1) is V25() real ext-real integer set
2 to_power ((c + 1) - 1) is V25() real ext-real Element of REAL
2 * (c + 1) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(2 * (c + 1)) ^2 is V25() real ext-real Element of REAL
(2 * (c + 1)) * (2 * (c + 1)) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer set
((2 * c) ^2) * 2 is V25() real ext-real Element of REAL
2 * 2 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
c * c is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer set
(c * c) + (2 * c) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
((c * c) + (2 * c)) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(2 * 2) * (((c * c) + (2 * c)) + 1) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(2 * 2) * (c * c) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
((2 * 2) * (c * c)) * 2 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(2 * c) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(2 * 2) * ((2 * c) + 1) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
((2 * 2) * (c * c)) + ((2 * 2) * ((2 * c) + 1)) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(((2 * 2) * (c * c)) * 2) - ((2 * 2) * (c * c)) is V25() real ext-real integer Element of REAL
- ((2 * 2) * (c * c)) is V25() real ext-real non positive integer set
(((2 * 2) * (c * c)) * 2) + (- ((2 * 2) * (c * c))) is V25() real ext-real integer set
(2 * 2) " is V25() real ext-real non negative Element of REAL
((2 * 2) ") * ((2 * 2) * (c * c)) is V25() real ext-real non negative Element of REAL
((2 * 2) ") * ((2 * 2) * ((2 * c) + 1)) is V25() real ext-real non negative Element of REAL
c ^2 is V25() real ext-real set
(c + 1) - 1 is V25() real ext-real integer Element of REAL
2 to_power ((c + 1) - 1) is V25() real ext-real Element of REAL
c + (- 1) is V25() real ext-real integer Element of REAL
(c + (- 1)) + 1 is V25() real ext-real integer Element of REAL
2 to_power ((c + (- 1)) + 1) is V25() real ext-real Element of REAL
2 to_power 1 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 |^ 1 is set
(2 to_power (c - 1)) * (2 to_power 1) is V25() real ext-real Element of REAL
(2 to_power (c - 1)) * 2 is V25() real ext-real Element of REAL
10 - 1 is V25() real ext-real integer Element of REAL
- 1 is V25() real ext-real non positive integer set
10 + (- 1) is V25() real ext-real integer set
2 to_power (10 - 1) is V25() real ext-real Element of REAL
6 + 3 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 to_power (6 + 3) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 |^ (6 + 3) is set
2 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 to_power (2 + 1) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 |^ (2 + 1) is set
64 * (2 to_power (2 + 1)) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 to_power 2 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 |^ 2 is set
2 to_power 1 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 |^ 1 is set
(2 to_power 2) * (2 to_power 1) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
64 * ((2 to_power 2) * (2 to_power 1)) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 to_power (1 + 1) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 |^ (1 + 1) is set
(2 to_power (1 + 1)) * 2 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
64 * ((2 to_power (1 + 1)) * 2) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(2 to_power 1) * (2 to_power 1) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
((2 to_power 1) * (2 to_power 1)) * 2 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
64 * (((2 to_power 1) * (2 to_power 1)) * 2) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 * (2 to_power 1) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(2 * (2 to_power 1)) * 2 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
64 * ((2 * (2 to_power 1)) * 2) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 * 2 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(2 * 2) * 2 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
64 * ((2 * 2) * 2) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
512 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
10 - 1 is V25() real ext-real integer Element of REAL
2 to_power (10 - 1) is V25() real ext-real Element of REAL
2 * 10 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(2 * 10) ^2 is V25() real ext-real Element of REAL
(2 * 10) * (2 * 10) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer set
c is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
c - 1 is V25() real ext-real integer Element of REAL
c + (- 1) is V25() real ext-real integer set
2 to_power (c - 1) is V25() real ext-real Element of REAL
2 * c is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(2 * c) ^2 is V25() real ext-real Element of REAL
(2 * c) * (2 * c) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer set
9 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
c is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer set
c to_power 6 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
c |^ 6 is set
2 * (c to_power 6) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
c + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(c + 1) to_power 6 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(c + 1) |^ 6 is set
2 * ((c + 1) to_power 6) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(c + 1) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
((c + 1) + 1) to_power 6 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
((c + 1) + 1) |^ 6 is set
((c + 1) to_power 6) / (c to_power 6) is V25() real ext-real non negative Element of REAL
(c to_power 6) " is V25() real ext-real non negative set
((c + 1) to_power 6) * ((c to_power 6) ") is V25() real ext-real non negative set
(c + 1) / c is V25() real ext-real non negative Element of REAL
c " is V25() real ext-real non negative set
(c + 1) * (c ") is V25() real ext-real non negative set
((c + 1) / c) to_power 6 is V25() real ext-real Element of REAL
((c + 1) / c) |^ 6 is set
c + 2 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(c + 2) / (c + 1) is V25() real ext-real non negative Element of REAL
(c + 1) " is non empty V25() real ext-real positive non negative set
(c + 2) * ((c + 1) ") is V25() real ext-real non negative set
((c + 1) / c) * (c + 1) is V25() real ext-real non negative Element of REAL
((c + 2) / (c + 1)) * (c + 1) is V25() real ext-real non negative Element of REAL
(((c + 1) / c) * (c + 1)) * c is V25() real ext-real non negative Element of REAL
(c + 2) * c is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
((c + 1) / c) * c is V25() real ext-real non negative Element of REAL
(((c + 1) / c) * c) * (c + 1) is V25() real ext-real non negative Element of REAL
(c + 1) ^2 is V25() real ext-real Element of REAL
(c + 1) * (c + 1) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer set
c ^2 is V25() real ext-real set
c * c is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer set
2 * c is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(c ^2) + (2 * c) is V25() real ext-real Element of REAL
1 * 1 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
((c ^2) + (2 * c)) + (1 * 1) is V25() real ext-real Element of REAL
((c ^2) + (2 * c)) - ((c ^2) + (2 * c)) is V25() real ext-real Element of REAL
- ((c ^2) + (2 * c)) is V25() real ext-real set
((c ^2) + (2 * c)) + (- ((c ^2) + (2 * c))) is V25() real ext-real set
(c + 1) " is non empty V25() real ext-real positive non negative Element of REAL
(c + 2) * ((c + 1) ") is V25() real ext-real non negative Element of REAL
0 * ((c + 1) ") is functional empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V25() real ext-real non positive non negative integer V49() V50() V51() V52() V53() V54() V55() FinSequence-membered Element of REAL
((c + 2) / (c + 1)) to_power 6 is V25() real ext-real Element of REAL
((c + 2) / (c + 1)) |^ 6 is set
(c + 2) to_power 6 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(c + 2) |^ 6 is set
((c + 2) to_power 6) / ((c + 1) to_power 6) is V25() real ext-real non negative Element of REAL
((c + 1) to_power 6) " is V25() real ext-real non negative set
((c + 2) to_power 6) * (((c + 1) to_power 6) ") is V25() real ext-real non negative set
(((c + 2) to_power 6) / ((c + 1) to_power 6)) * ((c + 1) to_power 6) is V25() real ext-real non negative Element of REAL
9 to_power 6 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
9 |^ 6 is set
2 * (9 to_power 6) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
9 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(9 + 1) to_power 6 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(9 + 1) |^ 6 is set
9 to_power 2 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
9 |^ 2 is set
1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
9 to_power (1 + 1) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
9 |^ (1 + 1) is set
9 to_power 1 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
9 |^ 1 is set
(9 to_power 1) * (9 to_power 1) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
9 * (9 to_power 1) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
9 * 9 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
81 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
9 to_power 4 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
9 |^ 4 is set
2 * (9 to_power 4) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 + 2 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
9 to_power (2 + 2) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
9 |^ (2 + 2) is set
2 * (9 to_power (2 + 2)) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
81 * 81 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 * (81 * 81) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
13122 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
13122 * 9 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(13122 * 9) * 9 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(9 to_power 4) * 9 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 * ((9 to_power 4) * 9) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(2 * ((9 to_power 4) * 9)) * 9 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(9 to_power 4) * (9 to_power 1) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 * ((9 to_power 4) * (9 to_power 1)) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(2 * ((9 to_power 4) * (9 to_power 1))) * 9 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
4 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
9 to_power (4 + 1) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
9 |^ (4 + 1) is set
2 * (9 to_power (4 + 1)) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(2 * (9 to_power (4 + 1))) * 9 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
9 to_power 5 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
9 |^ 5 is set
(9 to_power 5) * 9 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 * ((9 to_power 5) * 9) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(9 to_power 5) * (9 to_power 1) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 * ((9 to_power 5) * (9 to_power 1)) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
5 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
9 to_power (5 + 1) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
9 |^ (5 + 1) is set
2 * (9 to_power (5 + 1)) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
10 * 10 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(10 * 10) * 10 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
((10 * 10) * 10) * 10 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(((10 * 10) * 10) * 10) * 10 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
((((10 * 10) * 10) * 10) * 10) * 10 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
c is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
10 to_power (2 + 1) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
10 |^ (2 + 1) is set
10 to_power 2 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
10 |^ 2 is set
10 to_power 1 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
10 |^ 1 is set
(10 to_power 2) * (10 to_power 1) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
10 to_power (1 + 1) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
10 |^ (1 + 1) is set
(10 to_power (1 + 1)) * 10 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(10 to_power 1) * (10 to_power 1) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
((10 to_power 1) * (10 to_power 1)) * 10 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
10 * (10 to_power 1) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(10 * (10 to_power 1)) * 10 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
3 + 3 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
10 to_power (3 + 3) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
10 |^ (3 + 3) is set
((10 * 10) * 10) * ((10 * 10) * 10) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
c is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer set
c to_power 6 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
c |^ 6 is set
2 * (c to_power 6) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
c + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(c + 1) to_power 6 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(c + 1) |^ 6 is set
30 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
c is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer set
2 to_power c is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 |^ c is set
c to_power 6 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
c |^ 6 is set
c + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 to_power (c + 1) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 |^ (c + 1) is set
(c + 1) to_power 6 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(c + 1) |^ 6 is set
2 * (c to_power 6) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 to_power 1 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 |^ 1 is set
(2 to_power c) * (2 to_power 1) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(2 to_power c) * 2 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(c to_power 6) * 2 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 to_power 30 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 |^ 30 is set
5 * 6 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 to_power (5 * 6) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 |^ (5 * 6) is set
32 to_power 6 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
32 |^ 6 is set
30 to_power 6 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
30 |^ 6 is set
c is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 to_power c is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 |^ c is set
c to_power 6 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
c |^ 6 is set
c is V25() real ext-real Element of REAL
2 to_power c is V25() real ext-real Element of REAL
2 * c is V25() real ext-real Element of REAL
(2 * c) ^2 is V25() real ext-real Element of REAL
(2 * c) * (2 * c) is V25() real ext-real set
[/c\] is V25() real ext-real integer set
f is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 * f is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(2 * c) * (2 * c) is V25() real ext-real Element of REAL
(2 * f) ^2 is V25() real ext-real Element of REAL
(2 * f) * (2 * f) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer set
9 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
f - 1 is V25() real ext-real integer Element of REAL
- 1 is V25() real ext-real non positive integer set
f + (- 1) is V25() real ext-real integer set
2 to_power (f - 1) is V25() real ext-real Element of REAL
[\c/] is V25() real ext-real integer set
[/c\] - [\c/] is V25() real ext-real integer set
- [\c/] is V25() real ext-real integer set
[/c\] + (- [\c/]) is V25() real ext-real integer set
[\c/] + 1 is V25() real ext-real integer Element of REAL
1 + [\c/] is V25() real ext-real integer Element of REAL
2 to_power [\c/] is V25() real ext-real set
2 to_power 10 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 |^ 10 is set
c is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
f is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
log (2,f) is V25() real ext-real Element of REAL
2 to_power 9 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 |^ 9 is set
x is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 to_power 0 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 |^ 0 is set
log (2,1) is V25() real ext-real Element of REAL
(log (2,f)) * (log (2,f)) is V25() real ext-real Element of REAL
0 * (log (2,f)) is V25() real ext-real Element of REAL
4 * ((log (2,f)) * (log (2,f))) is V25() real ext-real Element of REAL
2 * ((log (2,f)) * (log (2,f))) is V25() real ext-real Element of REAL
2 to_power 1 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 |^ 1 is set
2 * f is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 * 2 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
f * f is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
f ^2 is V25() real ext-real Element of REAL
f * f is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer set
log (2,(f ^2)) is V25() real ext-real Element of REAL
2 ^2 is V25() real ext-real Element of REAL
2 * 2 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer set
log (2,(2 ^2)) is V25() real ext-real Element of REAL
2 to_power 2 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 |^ 2 is set
log (2,(2 to_power 2)) is V25() real ext-real Element of REAL
log (2,2) is V25() real ext-real Element of REAL
2 * (log (2,2)) is V25() real ext-real Element of REAL
2 * 1 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(log (2,(f ^2))) ^2 is V25() real ext-real Element of REAL
(log (2,(f ^2))) * (log (2,(f ^2))) is V25() real ext-real set
log (2,(2 to_power 9)) is V25() real ext-real Element of REAL
9 * (log (2,2)) is V25() real ext-real Element of REAL
9 * 1 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 * (log (2,f)) is V25() real ext-real Element of REAL
(2 * (log (2,f))) * (2 * (log (2,f))) is V25() real ext-real Element of REAL
0 * (2 * (log (2,f))) is V25() real ext-real Element of REAL
2 to_power (log (2,f)) is V25() real ext-real Element of REAL
log (2,(2 to_power (log (2,f)))) is V25() real ext-real Element of REAL
(2 * (log (2,f))) ^2 is V25() real ext-real Element of REAL
(2 * (log (2,f))) * (2 * (log (2,f))) is V25() real ext-real set
log (2,((2 * (log (2,f))) ^2)) is V25() real ext-real Element of REAL
(log (2,f)) * (log (2,2)) is V25() real ext-real Element of REAL
(log (2,f)) * 1 is V25() real ext-real Element of REAL
(2 * (log (2,f))) to_power 2 is V25() real ext-real Element of REAL
(2 * (log (2,f))) |^ 2 is set
log (2,((2 * (log (2,f))) to_power 2)) is V25() real ext-real Element of REAL
log (2,(2 * (log (2,f)))) is V25() real ext-real Element of REAL
2 * (log (2,(2 * (log (2,f))))) is V25() real ext-real Element of REAL
f to_power 2 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
f |^ 2 is set
log (2,(f to_power 2)) is V25() real ext-real Element of REAL
log (2,(log (2,(f to_power 2)))) is V25() real ext-real Element of REAL
2 * (log (2,(log (2,(f to_power 2))))) is V25() real ext-real Element of REAL
log (2,(log (2,(f ^2)))) is V25() real ext-real Element of REAL
2 * (log (2,(log (2,(f ^2))))) is V25() real ext-real Element of REAL
2 to_power (2 * (log (2,(log (2,(f ^2)))))) is V25() real ext-real Element of REAL
(log (2,(f ^2))) to_power 2 is V25() real ext-real Element of REAL
(log (2,(f ^2))) |^ 2 is set
log (2,((log (2,(f ^2))) to_power 2)) is V25() real ext-real Element of REAL
2 to_power (log (2,((log (2,(f ^2))) to_power 2))) is V25() real ext-real Element of REAL
log (2,((log (2,(f ^2))) ^2)) is V25() real ext-real Element of REAL
2 to_power (log (2,((log (2,(f ^2))) ^2))) is V25() real ext-real Element of REAL
(log (2,(f to_power 2))) ^2 is V25() real ext-real Element of REAL
(log (2,(f to_power 2))) * (log (2,(f to_power 2))) is V25() real ext-real set
f / 2 is V25() real ext-real non negative Element of REAL
f * (2 ") is V25() real ext-real non negative set
x is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
c is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
c is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
x is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
f is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
f to_power 6 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
f |^ 6 is set
2 to_power f is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 |^ f is set
log (2,(2 to_power f)) is V25() real ext-real Element of REAL
log (2,(f to_power 6)) is V25() real ext-real Element of REAL
log (2,2) is V25() real ext-real Element of REAL
f * (log (2,2)) is V25() real ext-real Element of REAL
f * 1 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
3 * 2 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
log (2,f) is V25() real ext-real Element of REAL
(3 * 2) * (log (2,f)) is V25() real ext-real Element of REAL
2 * (log (2,f)) is V25() real ext-real Element of REAL
3 * (2 * (log (2,f))) is V25() real ext-real Element of REAL
f / 3 is V25() real ext-real non negative Element of REAL
3 " is non empty V25() real ext-real positive non negative set
f * (3 ") is V25() real ext-real non negative set
x is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
x is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
7 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
f is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
e is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
e / 6 is V25() real ext-real non negative Element of REAL
6 " is non empty V25() real ext-real positive non negative set
e * (6 ") is V25() real ext-real non negative set
6 / 6 is V25() real ext-real non negative Element of REAL
6 * (6 ") is V25() real ext-real non negative set
f is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
f is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
max (f,2) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
max (x,c) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
max ((max (f,2)),(max (x,c))) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
N0 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
log (2,N0) is V25() real ext-real Element of REAL
2 * (log (2,N0)) is V25() real ext-real Element of REAL
1 + (2 * (log (2,N0))) is V25() real ext-real Element of REAL
(log (2,N0)) * (log (2,N0)) is V25() real ext-real Element of REAL
(1 + (2 * (log (2,N0)))) + ((log (2,N0)) * (log (2,N0))) is V25() real ext-real Element of REAL
1 + (log (2,N0)) is V25() real ext-real Element of REAL
(1 + (log (2,N0))) ^2 is V25() real ext-real Element of REAL
(1 + (log (2,N0))) * (1 + (log (2,N0))) is V25() real ext-real set
N0 / 3 is V25() real ext-real non negative Element of REAL
N0 * (3 ") is V25() real ext-real non negative set
N0 / 6 is V25() real ext-real non negative Element of REAL
N0 * (6 ") is V25() real ext-real non negative set
(N0 / 6) + (N0 / 3) is V25() real ext-real non negative Element of REAL
N0 / 2 is V25() real ext-real non negative Element of REAL
N0 * (2 ") is V25() real ext-real non negative set
((N0 / 6) + (N0 / 3)) + (N0 / 2) is V25() real ext-real non negative Element of REAL
sqrt N0 is V25() real ext-real Element of REAL
sqrt ((1 + (log (2,N0))) ^2) is V25() real ext-real Element of REAL
(sqrt N0) - (log (2,N0)) is V25() real ext-real Element of REAL
- (log (2,N0)) is V25() real ext-real set
(sqrt N0) + (- (log (2,N0))) is V25() real ext-real set
5 ! is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
120 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
4 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(4 + 1) ! is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
4 ! is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(4 + 1) * (4 !) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
3 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
3 ! is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(3 + 1) * (3 !) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
5 * ((3 + 1) * (3 !)) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 ! is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(2 + 1) * (2 !) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
4 * ((2 + 1) * (2 !)) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
5 * (4 * ((2 + 1) * (2 !))) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
14175 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
4096 / 14175 is V25() real ext-real non negative Element of REAL
14175 " is non empty V25() real ext-real positive non negative set
4096 * (14175 ") is V25() real ext-real non negative set
7 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
8 - 1 is V25() real ext-real integer Element of REAL
- 1 is V25() real ext-real non positive integer set
8 + (- 1) is V25() real ext-real integer set
c is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer set
c - 9 is V25() real ext-real integer Element of REAL
- 9 is V25() real ext-real non positive integer set
c + (- 9) is V25() real ext-real integer set
2 to_power (c - 9) is V25() real ext-real Element of REAL
1 / (2 to_power (c - 9)) is V25() real ext-real Element of REAL
(2 to_power (c - 9)) " is V25() real ext-real set
1 * ((2 to_power (c - 9)) ") is V25() real ext-real set
2 * c is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 to_power (2 * c) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 |^ (2 * c) is set
c ! is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(2 to_power (2 * c)) / (c !) is V25() real ext-real non negative Element of REAL
(c !) " is V25() real ext-real non negative set
(2 to_power (2 * c)) * ((c !) ") is V25() real ext-real non negative set
c + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(c + 1) - 9 is V25() real ext-real integer Element of REAL
(c + 1) + (- 9) is V25() real ext-real integer set
2 to_power ((c + 1) - 9) is V25() real ext-real Element of REAL
1 / (2 to_power ((c + 1) - 9)) is V25() real ext-real Element of REAL
(2 to_power ((c + 1) - 9)) " is V25() real ext-real set
1 * ((2 to_power ((c + 1) - 9)) ") is V25() real ext-real set
2 * (c + 1) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 to_power (2 * (c + 1)) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 |^ (2 * (c + 1)) is set
(c + 1) ! is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(2 to_power (2 * (c + 1))) / ((c + 1) !) is V25() real ext-real non negative Element of REAL
((c + 1) !) " is V25() real ext-real non negative set
(2 to_power (2 * (c + 1))) * (((c + 1) !) ") is V25() real ext-real non negative set
2 to_power 1 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 |^ 1 is set
1 / (2 to_power 1) is V25() real ext-real non negative Element of REAL
(2 to_power 1) " is V25() real ext-real non negative set
1 * ((2 to_power 1) ") is V25() real ext-real non negative set
2 to_power 2 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 |^ 2 is set
(2 to_power 2) / (c + 1) is V25() real ext-real non negative Element of REAL
(c + 1) " is non empty V25() real ext-real positive non negative set
(2 to_power 2) * ((c + 1) ") is V25() real ext-real non negative set
(1 / (2 to_power 1)) * (2 to_power 1) is V25() real ext-real non negative Element of REAL
(c + 1) " is non empty V25() real ext-real positive non negative Element of REAL
(2 to_power 2) * ((c + 1) ") is V25() real ext-real non negative Element of REAL
(2 to_power 1) * ((2 to_power 2) * ((c + 1) ")) is V25() real ext-real non negative Element of REAL
(2 to_power 1) * (2 to_power 2) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
((2 to_power 1) * (2 to_power 2)) * ((c + 1) ") is V25() real ext-real non negative Element of REAL
1 + 2 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 to_power (1 + 2) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 |^ (1 + 2) is set
(2 to_power (1 + 2)) * ((c + 1) ") is V25() real ext-real non negative Element of REAL
8 / (c + 1) is V25() real ext-real non negative Element of REAL
8 * ((c + 1) ") is V25() real ext-real non negative set
1 * (c + 1) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(8 / (c + 1)) * (c + 1) is V25() real ext-real non negative Element of REAL
- (c - 9) is V25() real ext-real integer Element of REAL
2 to_power (- (c - 9)) is V25() real ext-real Element of REAL
(1 / (2 to_power 1)) * (1 / (2 to_power (c - 9))) is V25() real ext-real Element of REAL
((2 to_power 2) / (c + 1)) * (1 / (2 to_power (c - 9))) is V25() real ext-real Element of REAL
0 * ((c + 1) ") is functional empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V25() real ext-real non positive non negative integer V49() V50() V51() V52() V53() V54() V55() FinSequence-membered Element of REAL
2 * 1 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(2 * c) + (2 * 1) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 to_power ((2 * c) + (2 * 1)) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 |^ ((2 * c) + (2 * 1)) is set
(2 to_power ((2 * c) + (2 * 1))) / ((c + 1) !) is V25() real ext-real non negative Element of REAL
(2 to_power ((2 * c) + (2 * 1))) * (((c + 1) !) ") is V25() real ext-real non negative set
(2 to_power (2 * c)) * (2 to_power 2) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
((2 to_power (2 * c)) * (2 to_power 2)) / ((c + 1) !) is V25() real ext-real non negative Element of REAL
((2 to_power (2 * c)) * (2 to_power 2)) * (((c + 1) !) ") is V25() real ext-real non negative set
(c + 1) * (c !) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
((2 to_power (2 * c)) * (2 to_power 2)) / ((c + 1) * (c !)) is V25() real ext-real non negative Element of REAL
((c + 1) * (c !)) " is V25() real ext-real non negative set
((2 to_power (2 * c)) * (2 to_power 2)) * (((c + 1) * (c !)) ") is V25() real ext-real non negative set
((2 to_power 2) / (c + 1)) * ((2 to_power (2 * c)) / (c !)) is V25() real ext-real non negative Element of REAL
(2 to_power 1) * (2 to_power (c - 9)) is V25() real ext-real Element of REAL
1 / ((2 to_power 1) * (2 to_power (c - 9))) is V25() real ext-real Element of REAL
((2 to_power 1) * (2 to_power (c - 9))) " is V25() real ext-real set
1 * (((2 to_power 1) * (2 to_power (c - 9))) ") is V25() real ext-real set
- 9 is V25() real ext-real non positive integer Element of REAL
c + (- 9) is V25() real ext-real integer Element of REAL
1 + (c + (- 9)) is V25() real ext-real integer Element of REAL
2 to_power (1 + (c + (- 9))) is V25() real ext-real Element of REAL
1 / (2 to_power (1 + (c + (- 9)))) is V25() real ext-real Element of REAL
(2 to_power (1 + (c + (- 9)))) " is V25() real ext-real set
1 * ((2 to_power (1 + (c + (- 9)))) ") is V25() real ext-real set
(c + 1) - 9 is V25() real ext-real integer Element of REAL
2 to_power ((c + 1) - 9) is V25() real ext-real Element of REAL
1 / (2 to_power ((c + 1) - 9)) is V25() real ext-real Element of REAL
(2 to_power ((c + 1) - 9)) " is V25() real ext-real set
1 * ((2 to_power ((c + 1) - 9)) ") is V25() real ext-real set
2 * 10 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 to_power (2 * 10) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 |^ (2 * 10) is set
10 ! is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(2 to_power (2 * 10)) / (10 !) is V25() real ext-real non negative Element of REAL
(10 !) " is V25() real ext-real non negative set
(2 to_power (2 * 10)) * ((10 !) ") is V25() real ext-real non negative set
20 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 to_power 20 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 |^ 20 is set
9 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
9 ! is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(9 + 1) * (9 !) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(2 to_power 20) / ((9 + 1) * (9 !)) is V25() real ext-real non negative Element of REAL
((9 + 1) * (9 !)) " is V25() real ext-real non negative set
(2 to_power 20) * (((9 + 1) * (9 !)) ") is V25() real ext-real non negative set
19 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
1 + 19 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 to_power (1 + 19) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 |^ (1 + 19) is set
10 * (9 !) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(2 to_power (1 + 19)) / (10 * (9 !)) is V25() real ext-real non negative Element of REAL
(10 * (9 !)) " is V25() real ext-real non negative set
(2 to_power (1 + 19)) * ((10 * (9 !)) ") is V25() real ext-real non negative set
2 to_power 1 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 |^ 1 is set
2 to_power 19 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 |^ 19 is set
(2 to_power 1) * (2 to_power 19) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
((2 to_power 1) * (2 to_power 19)) / (10 * (9 !)) is V25() real ext-real non negative Element of REAL
((2 to_power 1) * (2 to_power 19)) * ((10 * (9 !)) ") is V25() real ext-real non negative set
2 * (2 to_power 19) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
5 * (9 !) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 * (5 * (9 !)) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(2 * (2 to_power 19)) / (2 * (5 * (9 !))) is V25() real ext-real non negative Element of REAL
(2 * (5 * (9 !))) " is V25() real ext-real non negative set
(2 * (2 to_power 19)) * ((2 * (5 * (9 !))) ") is V25() real ext-real non negative set
(2 to_power 19) / (5 * (9 !)) is V25() real ext-real non negative Element of REAL
(5 * (9 !)) " is V25() real ext-real non negative set
(2 to_power 19) * ((5 * (9 !)) ") is V25() real ext-real non negative set
8 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
8 ! is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(8 + 1) * (8 !) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
5 * ((8 + 1) * (8 !)) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(2 to_power 19) / (5 * ((8 + 1) * (8 !))) is V25() real ext-real non negative Element of REAL
(5 * ((8 + 1) * (8 !))) " is V25() real ext-real non negative set
(2 to_power 19) * ((5 * ((8 + 1) * (8 !))) ") is V25() real ext-real non negative set
5 * 9 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(5 * 9) * (8 !) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(2 to_power 19) / ((5 * 9) * (8 !)) is V25() real ext-real non negative Element of REAL
((5 * 9) * (8 !)) " is V25() real ext-real non negative set
(2 to_power 19) * (((5 * 9) * (8 !)) ") is V25() real ext-real non negative set
45 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
7 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
7 ! is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(7 + 1) * (7 !) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
45 * ((7 + 1) * (7 !)) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(2 to_power 19) / (45 * ((7 + 1) * (7 !))) is V25() real ext-real non negative Element of REAL
(45 * ((7 + 1) * (7 !))) " is V25() real ext-real non negative set
(2 to_power 19) * ((45 * ((7 + 1) * (7 !))) ") is V25() real ext-real non negative set
3 + 16 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 to_power (3 + 16) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 |^ (3 + 16) is set
45 * (7 !) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
8 * (45 * (7 !)) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(2 to_power (3 + 16)) / (8 * (45 * (7 !))) is V25() real ext-real non negative Element of REAL
(8 * (45 * (7 !))) " is V25() real ext-real non negative set
(2 to_power (3 + 16)) * ((8 * (45 * (7 !))) ") is V25() real ext-real non negative set
2 to_power 16 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 |^ 16 is set
8 * (2 to_power 16) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(8 * (2 to_power 16)) / (8 * (45 * (7 !))) is V25() real ext-real non negative Element of REAL
(8 * (2 to_power 16)) * ((8 * (45 * (7 !))) ") is V25() real ext-real non negative set
(2 to_power 16) / (45 * (7 !)) is V25() real ext-real non negative Element of REAL
(45 * (7 !)) " is V25() real ext-real non negative set
(2 to_power 16) * ((45 * (7 !)) ") is V25() real ext-real non negative set
4 + 12 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 to_power (4 + 12) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 |^ (4 + 12) is set
6 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
6 ! is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(6 + 1) * (6 !) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
45 * ((6 + 1) * (6 !)) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(2 to_power (4 + 12)) / (45 * ((6 + 1) * (6 !))) is V25() real ext-real non negative Element of REAL
(45 * ((6 + 1) * (6 !))) " is V25() real ext-real non negative set
(2 to_power (4 + 12)) * ((45 * ((6 + 1) * (6 !))) ") is V25() real ext-real non negative set
3 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 to_power (3 + 1) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 |^ (3 + 1) is set
(2 to_power (3 + 1)) * 4096 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
((2 to_power (3 + 1)) * 4096) / (45 * ((6 + 1) * (6 !))) is V25() real ext-real non negative Element of REAL
((2 to_power (3 + 1)) * 4096) * ((45 * ((6 + 1) * (6 !))) ") is V25() real ext-real non negative set
8 * (2 to_power 1) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(8 * (2 to_power 1)) * 4096 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
((8 * (2 to_power 1)) * 4096) / (45 * ((6 + 1) * (6 !))) is V25() real ext-real non negative Element of REAL
((8 * (2 to_power 1)) * 4096) * ((45 * ((6 + 1) * (6 !))) ") is V25() real ext-real non negative set
8 * 2 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(8 * 2) * 4096 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
((8 * 2) * 4096) / (45 * ((6 + 1) * (6 !))) is V25() real ext-real non negative Element of REAL
((8 * 2) * 4096) * ((45 * ((6 + 1) * (6 !))) ") is V25() real ext-real non negative set
16 * 4096 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
45 * 7 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(45 * 7) * (6 !) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(16 * 4096) / ((45 * 7) * (6 !)) is V25() real ext-real non negative Element of REAL
((45 * 7) * (6 !)) " is V25() real ext-real non negative set
(16 * 4096) * (((45 * 7) * (6 !)) ") is V25() real ext-real non negative set
315 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
5 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(5 + 1) * (5 !) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
315 * ((5 + 1) * (5 !)) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(16 * 4096) / (315 * ((5 + 1) * (5 !))) is V25() real ext-real non negative Element of REAL
(315 * ((5 + 1) * (5 !))) " is V25() real ext-real non negative set
(16 * 4096) * ((315 * ((5 + 1) * (5 !))) ") is V25() real ext-real non negative set
10 - 9 is V25() real ext-real integer Element of REAL
- 9 is V25() real ext-real non positive integer set
10 + (- 9) is V25() real ext-real integer set
2 to_power (10 - 9) is V25() real ext-real Element of REAL
1 / (2 to_power (10 - 9)) is V25() real ext-real Element of REAL
(2 to_power (10 - 9)) " is V25() real ext-real set
1 * ((2 to_power (10 - 9)) ") is V25() real ext-real set
c is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
c - 9 is V25() real ext-real integer Element of REAL
c + (- 9) is V25() real ext-real integer set
2 to_power (c - 9) is V25() real ext-real Element of REAL
1 / (2 to_power (c - 9)) is V25() real ext-real Element of REAL
(2 to_power (c - 9)) " is V25() real ext-real set
1 * ((2 to_power (c - 9)) ") is V25() real ext-real set
2 * c is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 to_power (2 * c) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 |^ (2 * c) is set
c ! is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(2 to_power (2 * c)) / (c !) is V25() real ext-real non negative Element of REAL
(c !) " is V25() real ext-real non negative set
(2 to_power (2 * c)) * ((c !) ") is V25() real ext-real non negative set
c is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer set
c - 1 is V25() real ext-real integer Element of REAL
- 1 is V25() real ext-real non positive integer set
c + (- 1) is V25() real ext-real integer set
c - 2 is V25() real ext-real integer Element of REAL
- 2 is V25() real ext-real non positive integer set
c + (- 2) is V25() real ext-real integer set
2 * (c - 2) is V25() real ext-real integer Element of REAL
c + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(c + 1) - 1 is V25() real ext-real integer Element of REAL
(c + 1) + (- 1) is V25() real ext-real integer set
(c + 1) - 2 is V25() real ext-real integer Element of REAL
(c + 1) + (- 2) is V25() real ext-real integer set
2 * ((c + 1) - 2) is V25() real ext-real integer Element of REAL
c + (- 1) is V25() real ext-real integer Element of REAL
(c + (- 1)) + 1 is V25() real ext-real integer Element of REAL
(2 * (c - 2)) + 2 is V25() real ext-real integer Element of REAL
(c + 1) - 1 is V25() real ext-real integer Element of REAL
(c + 1) - 2 is V25() real ext-real integer Element of REAL
2 * ((c + 1) - 2) is V25() real ext-real integer Element of REAL
3 - 1 is V25() real ext-real integer Element of REAL
- 1 is V25() real ext-real non positive integer set
3 + (- 1) is V25() real ext-real integer set
3 - 2 is V25() real ext-real integer Element of REAL
- 2 is V25() real ext-real non positive integer set
3 + (- 2) is V25() real ext-real integer set
2 * (3 - 2) is V25() real ext-real integer Element of REAL
c is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
c - 1 is V25() real ext-real integer Element of REAL
c + (- 1) is V25() real ext-real integer set
c - 2 is V25() real ext-real integer Element of REAL
c + (- 2) is V25() real ext-real integer set
2 * (c - 2) is V25() real ext-real integer Element of REAL
5 to_power 5 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
5 |^ 5 is set
3125 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
4 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
5 to_power (4 + 1) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
5 |^ (4 + 1) is set
5 to_power 4 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
5 |^ 4 is set
5 to_power 1 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
5 |^ 1 is set
(5 to_power 4) * (5 to_power 1) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
3 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
5 to_power (3 + 1) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
5 |^ (3 + 1) is set
(5 to_power (3 + 1)) * 5 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
5 to_power 3 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
5 |^ 3 is set
(5 to_power 3) * (5 to_power 1) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
((5 to_power 3) * (5 to_power 1)) * 5 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
5 to_power (2 + 1) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
5 |^ (2 + 1) is set
(5 to_power (2 + 1)) * 5 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
((5 to_power (2 + 1)) * 5) * 5 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
5 to_power 2 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
5 |^ 2 is set
(5 to_power 2) * (5 to_power 1) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
((5 to_power 2) * (5 to_power 1)) * 5 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(((5 to_power 2) * (5 to_power 1)) * 5) * 5 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
5 to_power (1 + 1) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
5 |^ (1 + 1) is set
(5 to_power (1 + 1)) * 5 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
((5 to_power (1 + 1)) * 5) * 5 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(((5 to_power (1 + 1)) * 5) * 5) * 5 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(5 to_power 1) * (5 to_power 1) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
((5 to_power 1) * (5 to_power 1)) * 5 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(((5 to_power 1) * (5 to_power 1)) * 5) * 5 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
((((5 to_power 1) * (5 to_power 1)) * 5) * 5) * 5 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(5 to_power 1) * 5 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
((5 to_power 1) * 5) * 5 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(((5 to_power 1) * 5) * 5) * 5 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
((((5 to_power 1) * 5) * 5) * 5) * 5 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
5 * 5 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(5 * 5) * 5 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
((5 * 5) * 5) * 5 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(((5 * 5) * 5) * 5) * 5 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
4 to_power 4 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
4 |^ 4 is set
256 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
3 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
4 to_power (3 + 1) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
4 |^ (3 + 1) is set
4 to_power 3 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
4 |^ 3 is set
4 to_power 1 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
4 |^ 1 is set
(4 to_power 3) * (4 to_power 1) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
4 to_power (2 + 1) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
4 |^ (2 + 1) is set
(4 to_power (2 + 1)) * 4 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
4 to_power 2 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
4 |^ 2 is set
(4 to_power 2) * (4 to_power 1) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
((4 to_power 2) * (4 to_power 1)) * 4 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
4 to_power (1 + 1) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
4 |^ (1 + 1) is set
(4 to_power (1 + 1)) * 4 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
((4 to_power (1 + 1)) * 4) * 4 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(4 to_power 1) * (4 to_power 1) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
((4 to_power 1) * (4 to_power 1)) * 4 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(((4 to_power 1) * (4 to_power 1)) * 4) * 4 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(4 to_power 1) * 4 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
((4 to_power 1) * 4) * 4 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(((4 to_power 1) * 4) * 4) * 4 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
4 * 4 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(4 * 4) * 4 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
((4 * 4) * 4) * 4 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
c is V25() real ext-real Element of REAL
x is V25() real ext-real Element of REAL
c / x is V25() real ext-real Element of REAL
x " is V25() real ext-real set
c * (x ") is V25() real ext-real set
f is V25() real ext-real Element of REAL
e is V25() real ext-real Element of REAL
f / e is V25() real ext-real Element of REAL
e " is V25() real ext-real set
f * (e ") is V25() real ext-real set
(c / x) / (f / e) is V25() real ext-real Element of REAL
(f / e) " is V25() real ext-real set
(c / x) * ((f / e) ") is V25() real ext-real set
c / f is V25() real ext-real Element of REAL
f " is V25() real ext-real set
c * (f ") is V25() real ext-real set
e / x is V25() real ext-real Element of REAL
e * (x ") is V25() real ext-real set
(c / f) * (e / x) is V25() real ext-real Element of REAL
c * e is V25() real ext-real Element of REAL
x * f is V25() real ext-real Element of REAL
(c * e) / (x * f) is V25() real ext-real Element of REAL
(x * f) " is V25() real ext-real set
(c * e) * ((x * f) ") is V25() real ext-real set
c is V25() real ext-real set
sqrt c is V25() real ext-real set
c to_power (1 / 2) is V25() real ext-real set
(c to_power (1 / 2)) ^2 is V25() real ext-real set
(c to_power (1 / 2)) * (c to_power (1 / 2)) is V25() real ext-real set
(c to_power (1 / 2)) to_power 2 is V25() real ext-real set
(c to_power (1 / 2)) |^ 2 is set
(1 / 2) * 2 is V25() real ext-real non negative Element of REAL
c to_power ((1 / 2) * 2) is V25() real ext-real set
() /" ((1 / 2)) is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() Element of K19(K20(NAT,REAL))
((1 / 2)) " is V1() V4( NAT ) Function-like V14( NAT ) V35() V36() V37() set
() (#) (((1 / 2)) ") is V1() V4( NAT ) Function-like V14( NAT ) V35() V36() V37() set
lim (() /" ((1 / 2))) is V25() real ext-real Element of REAL
e is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
max (2,e) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
t is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
sqrt t is V25() real ext-real Element of REAL
(() /" ((1 / 2))) . t is V25() real ext-real Element of REAL
((() /" ((1 / 2))) . t) - 0 is V25() real ext-real Element of REAL
- 0 is functional empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V25() real ext-real non positive non negative integer V49() V50() V51() V52() V53() V54() V55() FinSequence-membered set
((() /" ((1 / 2))) . t) + (- 0) is V25() real ext-real set
abs (((() /" ((1 / 2))) . t) - 0) is V25() real ext-real Element of REAL
() . t is V25() real ext-real Element of REAL
((1 / 2)) . t is V25() real ext-real Element of REAL
(() . t) / (((1 / 2)) . t) is V25() real ext-real Element of REAL
(((1 / 2)) . t) " is V25() real ext-real set
(() . t) * ((((1 / 2)) . t) ") is V25() real ext-real set
log (2,t) is V25() real ext-real Element of REAL
(log (2,t)) / (((1 / 2)) . t) is V25() real ext-real Element of REAL
(log (2,t)) * ((((1 / 2)) . t) ") is V25() real ext-real set
t to_power (1 / 2) is V25() real ext-real Element of REAL
(log (2,t)) / (t to_power (1 / 2)) is V25() real ext-real Element of REAL
(t to_power (1 / 2)) " is V25() real ext-real set
(log (2,t)) * ((t to_power (1 / 2)) ") is V25() real ext-real set
(log (2,t)) / (sqrt t) is V25() real ext-real Element of REAL
(sqrt t) " is V25() real ext-real set
(log (2,t)) * ((sqrt t) ") is V25() real ext-real set
(sqrt t) * (1 / 2) is V25() real ext-real Element of REAL
((log (2,t)) / (sqrt t)) * (sqrt t) is V25() real ext-real Element of REAL
(sqrt t) * ((sqrt t) * (1 / 2)) is V25() real ext-real Element of REAL
(sqrt t) * (log (2,t)) is V25() real ext-real Element of REAL
(sqrt t) ^2 is V25() real ext-real Element of REAL
(sqrt t) * (sqrt t) is V25() real ext-real set
((sqrt t) ^2) * (1 / 2) is V25() real ext-real Element of REAL
t * (1 / 2) is V25() real ext-real non negative Element of REAL
t / 2 is V25() real ext-real non negative Element of REAL
t * (2 ") is V25() real ext-real non negative set
(t / 2) + (t / 2) is V25() real ext-real non negative Element of REAL
(t / 2) + ((sqrt t) * (log (2,t))) is V25() real ext-real Element of REAL
t - ((sqrt t) * (log (2,t))) is V25() real ext-real Element of REAL
- ((sqrt t) * (log (2,t))) is V25() real ext-real set
t + (- ((sqrt t) * (log (2,t)))) is V25() real ext-real set
c is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() Element of K19(K20(NAT,REAL))
x is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
x + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(x + 1) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
1 / ((x + 1) + 1) is V25() real ext-real non negative Element of REAL
((x + 1) + 1) " is non empty V25() real ext-real positive non negative set
1 * (((x + 1) + 1) ") is V25() real ext-real non negative set
1 + (1 / ((x + 1) + 1)) is non empty V25() real ext-real positive non negative Element of REAL
1 / (x + 1) is V25() real ext-real non negative Element of REAL
(x + 1) " is non empty V25() real ext-real positive non negative set
1 * ((x + 1) ") is V25() real ext-real non negative set
1 + (1 / (x + 1)) is non empty V25() real ext-real positive non negative Element of REAL
(1 + (1 / ((x + 1) + 1))) / (1 + (1 / (x + 1))) is V25() real ext-real non negative Element of REAL
(1 + (1 / (x + 1))) " is non empty V25() real ext-real positive non negative set
(1 + (1 / ((x + 1) + 1))) * ((1 + (1 / (x + 1))) ") is V25() real ext-real non negative set
1 * ((x + 1) + 1) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(1 * ((x + 1) + 1)) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
((1 * ((x + 1) + 1)) + 1) / ((x + 1) + 1) is V25() real ext-real non negative Element of REAL
((1 * ((x + 1) + 1)) + 1) * (((x + 1) + 1) ") is V25() real ext-real non negative set
(((1 * ((x + 1) + 1)) + 1) / ((x + 1) + 1)) / (1 + (1 / (x + 1))) is V25() real ext-real non negative Element of REAL
(((1 * ((x + 1) + 1)) + 1) / ((x + 1) + 1)) * ((1 + (1 / (x + 1))) ") is V25() real ext-real non negative set
((x + 1) + 1) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(((x + 1) + 1) + 1) / ((x + 1) + 1) is V25() real ext-real non negative Element of REAL
(((x + 1) + 1) + 1) * (((x + 1) + 1) ") is V25() real ext-real non negative set
1 * (x + 1) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(1 * (x + 1)) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
((1 * (x + 1)) + 1) / (x + 1) is V25() real ext-real non negative Element of REAL
((1 * (x + 1)) + 1) * ((x + 1) ") is V25() real ext-real non negative set
((((x + 1) + 1) + 1) / ((x + 1) + 1)) / (((1 * (x + 1)) + 1) / (x + 1)) is V25() real ext-real non negative Element of REAL
(((1 * (x + 1)) + 1) / (x + 1)) " is V25() real ext-real non negative set
((((x + 1) + 1) + 1) / ((x + 1) + 1)) * ((((1 * (x + 1)) + 1) / (x + 1)) ") is V25() real ext-real non negative set
1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
x + (1 + 1) is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(x + (1 + 1)) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
((x + (1 + 1)) + 1) * (x + 1) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
x + 2 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(x + 2) * (x + 2) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(((x + (1 + 1)) + 1) * (x + 1)) / ((x + 2) * (x + 2)) is V25() real ext-real non negative Element of REAL
((x + 2) * (x + 2)) " is V25() real ext-real non negative set
(((x + (1 + 1)) + 1) * (x + 1)) * (((x + 2) * (x + 2)) ") is V25() real ext-real non negative set
x * x is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
x * 2 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(x * x) + (x * 2) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 * x is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
((x * x) + (x * 2)) + (2 * x) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(((x * x) + (x * 2)) + (2 * x)) + 3 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
((((x * x) + (x * 2)) + (2 * x)) + 3) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(((((x * x) + (x * 2)) + (2 * x)) + 3) + 1) - 1 is V25() real ext-real integer Element of REAL
- 1 is V25() real ext-real non positive integer set
(((((x * x) + (x * 2)) + (2 * x)) + 3) + 1) + (- 1) is V25() real ext-real integer set
((((((x * x) + (x * 2)) + (2 * x)) + 3) + 1) - 1) / ((x + 2) * (x + 2)) is V25() real ext-real Element of REAL
((((((x * x) + (x * 2)) + (2 * x)) + 3) + 1) - 1) * (((x + 2) * (x + 2)) ") is V25() real ext-real set
((x + 2) * (x + 2)) / ((x + 2) * (x + 2)) is V25() real ext-real non negative Element of REAL
((x + 2) * (x + 2)) * (((x + 2) * (x + 2)) ") is V25() real ext-real non negative set
1 / ((x + 2) * (x + 2)) is V25() real ext-real non negative Element of REAL
1 * (((x + 2) * (x + 2)) ") is V25() real ext-real non negative set
(((x + 2) * (x + 2)) / ((x + 2) * (x + 2))) - (1 / ((x + 2) * (x + 2))) is V25() real ext-real Element of REAL
- (1 / ((x + 2) * (x + 2))) is V25() real ext-real non positive set
(((x + 2) * (x + 2)) / ((x + 2) * (x + 2))) + (- (1 / ((x + 2) * (x + 2)))) is V25() real ext-real set
1 - (1 / ((x + 2) * (x + 2))) is V25() real ext-real Element of REAL
1 + (- (1 / ((x + 2) * (x + 2)))) is V25() real ext-real set
0 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
- (1 / ((x + 2) * (x + 2))) is V25() real ext-real non positive Element of REAL
((x + 1) + 1) * (- (1 / ((x + 2) * (x + 2)))) is V25() real ext-real non positive Element of REAL
1 + (((x + 1) + 1) * (- (1 / ((x + 2) * (x + 2))))) is V25() real ext-real Element of REAL
1 + (- (1 / ((x + 2) * (x + 2)))) is V25() real ext-real Element of REAL
(1 + (- (1 / ((x + 2) * (x + 2))))) to_power ((x + 1) + 1) is V25() real ext-real Element of REAL
(1 + (- (1 / ((x + 2) * (x + 2))))) |^ ((x + 1) + 1) is set
(x + 2) * 1 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
((x + 2) * 1) / ((x + 2) * (x + 2)) is V25() real ext-real non negative Element of REAL
((x + 2) * 1) * (((x + 2) * (x + 2)) ") is V25() real ext-real non negative set
1 - (((x + 2) * 1) / ((x + 2) * (x + 2))) is V25() real ext-real Element of REAL
- (((x + 2) * 1) / ((x + 2) * (x + 2))) is V25() real ext-real non positive set
1 + (- (((x + 2) * 1) / ((x + 2) * (x + 2)))) is V25() real ext-real set
(1 - (1 / ((x + 2) * (x + 2)))) to_power ((x + 1) + 1) is V25() real ext-real Element of REAL
(1 - (1 / ((x + 2) * (x + 2)))) |^ ((x + 1) + 1) is set
(x + 2) / (x + 2) is V25() real ext-real non negative Element of REAL
(x + 2) " is non empty V25() real ext-real positive non negative set
(x + 2) * ((x + 2) ") is V25() real ext-real non negative set
((x + 2) / (x + 2)) * 1 is V25() real ext-real non negative Element of REAL
(((x + 2) / (x + 2)) * 1) / (x + 2) is V25() real ext-real non negative Element of REAL
(((x + 2) / (x + 2)) * 1) * ((x + 2) ") is V25() real ext-real non negative set
1 - ((((x + 2) / (x + 2)) * 1) / (x + 2)) is V25() real ext-real Element of REAL
- ((((x + 2) / (x + 2)) * 1) / (x + 2)) is V25() real ext-real non positive set
1 + (- ((((x + 2) / (x + 2)) * 1) / (x + 2))) is V25() real ext-real set
1 * 1 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(1 * 1) / (x + 2) is V25() real ext-real non negative Element of REAL
(1 * 1) * ((x + 2) ") is V25() real ext-real non negative set
1 - ((1 * 1) / (x + 2)) is V25() real ext-real Element of REAL
- ((1 * 1) / (x + 2)) is V25() real ext-real non positive set
1 + (- ((1 * 1) / (x + 2))) is V25() real ext-real set
c . (x + 1) is V25() real ext-real Element of REAL
c . x is V25() real ext-real Element of REAL
(c . (x + 1)) / (c . x) is V25() real ext-real Element of REAL
(c . x) " is V25() real ext-real set
(c . (x + 1)) * ((c . x) ") is V25() real ext-real set
(1 + (1 / ((x + 1) + 1))) to_power ((x + 1) + 1) is V25() real ext-real Element of REAL
(1 + (1 / ((x + 1) + 1))) |^ ((x + 1) + 1) is set
((1 + (1 / ((x + 1) + 1))) to_power ((x + 1) + 1)) / (c . x) is V25() real ext-real Element of REAL
((1 + (1 / ((x + 1) + 1))) to_power ((x + 1) + 1)) * ((c . x) ") is V25() real ext-real set
(1 + (1 / (x + 1))) to_power (x + 1) is V25() real ext-real Element of REAL
(1 + (1 / (x + 1))) |^ (x + 1) is set
((1 + (1 / ((x + 1) + 1))) to_power ((x + 1) + 1)) / ((1 + (1 / (x + 1))) to_power (x + 1)) is V25() real ext-real Element of REAL
((1 + (1 / (x + 1))) to_power (x + 1)) " is V25() real ext-real set
((1 + (1 / ((x + 1) + 1))) to_power ((x + 1) + 1)) * (((1 + (1 / (x + 1))) to_power (x + 1)) ") is V25() real ext-real set
(((1 + (1 / ((x + 1) + 1))) to_power ((x + 1) + 1)) / ((1 + (1 / (x + 1))) to_power (x + 1))) * 1 is V25() real ext-real Element of REAL
(1 + (1 / (x + 1))) / (1 + (1 / (x + 1))) is V25() real ext-real non negative Element of REAL
(1 + (1 / (x + 1))) * ((1 + (1 / (x + 1))) ") is V25() real ext-real non negative set
(((1 + (1 / ((x + 1) + 1))) to_power ((x + 1) + 1)) / ((1 + (1 / (x + 1))) to_power (x + 1))) * ((1 + (1 / (x + 1))) / (1 + (1 / (x + 1)))) is V25() real ext-real Element of REAL
(1 + (1 / (x + 1))) * ((1 + (1 / ((x + 1) + 1))) to_power ((x + 1) + 1)) is V25() real ext-real Element of REAL
((1 + (1 / (x + 1))) to_power (x + 1)) * (1 + (1 / (x + 1))) is V25() real ext-real Element of REAL
((1 + (1 / (x + 1))) * ((1 + (1 / ((x + 1) + 1))) to_power ((x + 1) + 1))) / (((1 + (1 / (x + 1))) to_power (x + 1)) * (1 + (1 / (x + 1)))) is V25() real ext-real Element of REAL
(((1 + (1 / (x + 1))) to_power (x + 1)) * (1 + (1 / (x + 1)))) " is V25() real ext-real set
((1 + (1 / (x + 1))) * ((1 + (1 / ((x + 1) + 1))) to_power ((x + 1) + 1))) * ((((1 + (1 / (x + 1))) to_power (x + 1)) * (1 + (1 / (x + 1)))) ") is V25() real ext-real set
(1 + (1 / (x + 1))) to_power 1 is V25() real ext-real Element of REAL
(1 + (1 / (x + 1))) |^ 1 is set
((1 + (1 / (x + 1))) to_power (x + 1)) * ((1 + (1 / (x + 1))) to_power 1) is V25() real ext-real Element of REAL
((1 + (1 / (x + 1))) * ((1 + (1 / ((x + 1) + 1))) to_power ((x + 1) + 1))) / (((1 + (1 / (x + 1))) to_power (x + 1)) * ((1 + (1 / (x + 1))) to_power 1)) is V25() real ext-real Element of REAL
(((1 + (1 / (x + 1))) to_power (x + 1)) * ((1 + (1 / (x + 1))) to_power 1)) " is V25() real ext-real set
((1 + (1 / (x + 1))) * ((1 + (1 / ((x + 1) + 1))) to_power ((x + 1) + 1))) * ((((1 + (1 / (x + 1))) to_power (x + 1)) * ((1 + (1 / (x + 1))) to_power 1)) ") is V25() real ext-real set
(1 + (1 / (x + 1))) to_power ((x + 1) + 1) is V25() real ext-real Element of REAL
(1 + (1 / (x + 1))) |^ ((x + 1) + 1) is set
((1 + (1 / (x + 1))) * ((1 + (1 / ((x + 1) + 1))) to_power ((x + 1) + 1))) / ((1 + (1 / (x + 1))) to_power ((x + 1) + 1)) is V25() real ext-real Element of REAL
((1 + (1 / (x + 1))) to_power ((x + 1) + 1)) " is V25() real ext-real set
((1 + (1 / (x + 1))) * ((1 + (1 / ((x + 1) + 1))) to_power ((x + 1) + 1))) * (((1 + (1 / (x + 1))) to_power ((x + 1) + 1)) ") is V25() real ext-real set
((1 + (1 / ((x + 1) + 1))) to_power ((x + 1) + 1)) / ((1 + (1 / (x + 1))) to_power ((x + 1) + 1)) is V25() real ext-real Element of REAL
((1 + (1 / ((x + 1) + 1))) to_power ((x + 1) + 1)) * (((1 + (1 / (x + 1))) to_power ((x + 1) + 1)) ") is V25() real ext-real set
(1 + (1 / (x + 1))) * (((1 + (1 / ((x + 1) + 1))) to_power ((x + 1) + 1)) / ((1 + (1 / (x + 1))) to_power ((x + 1) + 1))) is V25() real ext-real Element of REAL
((1 + (1 / ((x + 1) + 1))) / (1 + (1 / (x + 1)))) to_power ((x + 1) + 1) is V25() real ext-real Element of REAL
((1 + (1 / ((x + 1) + 1))) / (1 + (1 / (x + 1)))) |^ ((x + 1) + 1) is set
(1 + (1 / (x + 1))) * (((1 + (1 / ((x + 1) + 1))) / (1 + (1 / (x + 1)))) to_power ((x + 1) + 1)) is V25() real ext-real Element of REAL
1 / (x + 2) is V25() real ext-real non negative Element of REAL
1 * ((x + 2) ") is V25() real ext-real non negative set
1 - (1 / (x + 2)) is V25() real ext-real Element of REAL
- (1 / (x + 2)) is V25() real ext-real non positive set
1 + (- (1 / (x + 2))) is V25() real ext-real set
(1 + (1 / (x + 1))) * (1 - (1 / (x + 2))) is V25() real ext-real Element of REAL
(((1 * (x + 1)) + 1) / (x + 1)) * (1 - (1 / (x + 2))) is V25() real ext-real Element of REAL
(x + 2) / (x + 1) is V25() real ext-real non negative Element of REAL
(x + 2) * ((x + 1) ") is V25() real ext-real non negative set
1 * (x + 2) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(1 * (x + 2)) - 1 is V25() real ext-real integer Element of REAL
(1 * (x + 2)) + (- 1) is V25() real ext-real integer set
((1 * (x + 2)) - 1) / (x + 2) is V25() real ext-real Element of REAL
((1 * (x + 2)) - 1) * ((x + 2) ") is V25() real ext-real set
((x + 2) / (x + 1)) * (((1 * (x + 2)) - 1) / (x + 2)) is V25() real ext-real Element of REAL
(x + 1) * (x + 2) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
((x + 1) * (x + 2)) / ((x + 1) * (x + 2)) is V25() real ext-real non negative Element of REAL
((x + 1) * (x + 2)) " is V25() real ext-real non negative set
((x + 1) * (x + 2)) * (((x + 1) * (x + 2)) ") is V25() real ext-real non negative set
c is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
c + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(c + 1) / c is V25() real ext-real non negative Element of REAL
c " is V25() real ext-real non negative set
(c + 1) * (c ") is V25() real ext-real non negative set
((c + 1) / c) to_power c is V25() real ext-real Element of REAL
((c + 1) / c) |^ c is set
c + 2 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(c + 2) / (c + 1) is V25() real ext-real non negative Element of REAL
(c + 1) " is non empty V25() real ext-real positive non negative set
(c + 2) * ((c + 1) ") is V25() real ext-real non negative set
((c + 2) / (c + 1)) to_power (c + 1) is V25() real ext-real Element of REAL
((c + 2) / (c + 1)) |^ (c + 1) is set
x is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() Element of K19(K20(NAT,REAL))
0 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
c - 1 is V25() real ext-real integer Element of REAL
- 1 is V25() real ext-real non positive integer set
c + (- 1) is V25() real ext-real integer set
f is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
x . f is V25() real ext-real Element of REAL
f + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
x . (f + 1) is V25() real ext-real Element of REAL
1 / (f + 1) is V25() real ext-real non negative Element of REAL
(f + 1) " is non empty V25() real ext-real positive non negative set
1 * ((f + 1) ") is V25() real ext-real non negative set
1 + (1 / (f + 1)) is non empty V25() real ext-real positive non negative Element of REAL
(1 + (1 / (f + 1))) to_power (f + 1) is V25() real ext-real Element of REAL
(1 + (1 / (f + 1))) |^ (f + 1) is set
1 / c is V25() real ext-real non negative Element of REAL
1 * (c ") is V25() real ext-real non negative set
1 + (1 / c) is non empty V25() real ext-real positive non negative Element of REAL
(1 + (1 / c)) to_power c is V25() real ext-real Element of REAL
(1 + (1 / c)) |^ c is set
1 / (c + 1) is V25() real ext-real non negative Element of REAL
1 * ((c + 1) ") is V25() real ext-real non negative set
1 + (1 / (c + 1)) is non empty V25() real ext-real positive non negative Element of REAL
(1 + (1 / (c + 1))) to_power (c + 1) is V25() real ext-real Element of REAL
(1 + (1 / (c + 1))) |^ (c + 1) is set
c / c is V25() real ext-real non negative Element of REAL
c * (c ") is V25() real ext-real non negative set
(c / c) + (1 / c) is V25() real ext-real non negative Element of REAL
((c / c) + (1 / c)) to_power c is V25() real ext-real Element of REAL
((c / c) + (1 / c)) |^ c is set
(c + 1) / (c + 1) is V25() real ext-real non negative Element of REAL
(c + 1) * ((c + 1) ") is V25() real ext-real non negative set
((c + 1) / (c + 1)) + (1 / (c + 1)) is V25() real ext-real non negative Element of REAL
(((c + 1) / (c + 1)) + (1 / (c + 1))) to_power (c + 1) is V25() real ext-real Element of REAL
(((c + 1) / (c + 1)) + (1 / (c + 1))) |^ (c + 1) is set
c is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() Element of K19(K20(NAT,REAL))
x is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() Element of K19(K20(NAT,REAL))
c /" x is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() Element of K19(K20(NAT,REAL))
x " is V1() V4( NAT ) Function-like V14( NAT ) V35() V36() V37() set
c (#) (x ") is V1() V4( NAT ) Function-like V14( NAT ) V35() V36() V37() set
e is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
x . e is V25() real ext-real Element of REAL
sqrt e is V25() real ext-real Element of REAL
e to_power (sqrt e) is V25() real ext-real Element of REAL
N0 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
c . N0 is V25() real ext-real Element of REAL
log (2,N0) is V25() real ext-real Element of REAL
N0 to_power (log (2,N0)) is V25() real ext-real Element of REAL
N0 is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() eventually-nonnegative eventually-positive eventually-nonzero Element of K19(K20(NAT,REAL))
Big_Oh N0 is functional non empty FUNCTION_DOMAIN of NAT , REAL
{ b₁ where b₁ is V1() V4( NAT ) V5( REAL ) Function-like V14( NAT ) quasi_total V35() V36() V37() Element of Funcs (NAT,REAL) : ex b₂ being V25() real ext-real Element of REAL ex b₃ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT st
( not b₂ <= 0 & ( for b₄ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT holds
( not b₃ <= b₄ or ( b₁ . b₄ <= b₂ * (N0 . b₄) & 0 <= b₁ . b₄ ) ) ) ) } is set
e is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() eventually-nonnegative eventually-positive eventually-nonzero Element of K19(K20(NAT,REAL))
Big_Oh e is functional non empty FUNCTION_DOMAIN of NAT , REAL
{ b₁ where b₁ is V1() V4( NAT ) V5( REAL ) Function-like V14( NAT ) quasi_total V35() V36() V37() Element of Funcs (NAT,REAL) : ex b₂ being V25() real ext-real Element of REAL ex b₃ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT st
( not b₂ <= 0 & ( for b₄ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT holds
( not b₃ <= b₄ or ( b₁ . b₄ <= b₂ * (e . b₄) & 0 <= b₁ . b₄ ) ) ) ) } is set
t is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
t is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
d is V25() real ext-real set
1 / d is V25() real ext-real Element of REAL
d " is V25() real ext-real set
1 * (d ") is V25() real ext-real set
[/(1 / d)\] is V25() real ext-real integer set
max ([/(1 / d)\],2) is V25() real ext-real set
max (t,(max ([/(1 / d)\],2))) is V25() real ext-real set
b is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
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
(c /" x) . n is V25() real ext-real Element of REAL
N0 . n is V25() real ext-real Element of REAL
e . n is V25() real ext-real Element of REAL
(N0 . n) / (e . n) is V25() real ext-real Element of REAL
(e . n) " is V25() real ext-real set
(N0 . n) * ((e . n) ") is V25() real ext-real set
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
log (2,n) is V25() real ext-real Element of REAL
n to_power (log (2,n)) is V25() real ext-real Element of REAL
sqrt n is V25() real ext-real Element of REAL
n to_power (sqrt n) is V25() real ext-real Element of REAL
(n to_power (log (2,n))) / (n to_power (sqrt n)) is V25() real ext-real Element of REAL
(n to_power (sqrt n)) " is V25() real ext-real set
(n to_power (log (2,n))) * ((n to_power (sqrt n)) ") is V25() real ext-real set
(log (2,n)) - (sqrt n) is V25() real ext-real Element of REAL
- (sqrt n) is V25() real ext-real set
(log (2,n)) + (- (sqrt n)) is V25() real ext-real set
n to_power ((log (2,n)) - (sqrt n)) is V25() real ext-real Element of REAL
(sqrt n) - (log (2,n)) is V25() real ext-real Element of REAL
- (log (2,n)) is V25() real ext-real set
(sqrt n) + (- (log (2,n))) is V25() real ext-real set
- ((sqrt n) - (log (2,n))) is V25() real ext-real Element of REAL
n to_power (- ((sqrt n) - (log (2,n)))) is V25() real ext-real Element of REAL
(- 1) * 1 is V25() real ext-real non positive integer Element of REAL
(- 1) * ((sqrt n) - (log (2,n))) is V25() real ext-real Element of REAL
n to_power (- 1) is V25() real ext-real Element of REAL
1 / n is V25() real ext-real non negative Element of REAL
n " is V25() real ext-real non negative set
1 * (n ") is V25() real ext-real non negative set
1 / (1 / d) is V25() real ext-real Element of REAL
(1 / d) " is V25() real ext-real set
1 * ((1 / d) ") is V25() real ext-real set
n to_power 1 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
n |^ 1 is set
1 / (n to_power 1) is V25() real ext-real non negative Element of REAL
(n to_power 1) " is V25() real ext-real non negative set
1 * ((n to_power 1) ") is V25() real ext-real non negative set
((c /" x) . n) - 0 is V25() real ext-real Element of REAL
- 0 is functional empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V25() real ext-real non positive non negative integer V49() V50() V51() V52() V53() V54() V55() FinSequence-membered set
((c /" x) . n) + (- 0) is V25() real ext-real set
abs (((c /" x) . n) - 0) is V25() real ext-real Element of REAL
lim (c /" x) is V25() real ext-real Element of REAL
Big_Omega e is functional non empty FUNCTION_DOMAIN of NAT , REAL
{ b₁ where b₁ is V1() V4( NAT ) V5( REAL ) Function-like V14( NAT ) quasi_total V35() V36() V37() Element of Funcs (NAT,REAL) : ex b₂ being V25() real ext-real Element of REAL ex b₃ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT st
( not b₂ <= 0 & ( for b₄ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT holds
( not b₃ <= b₄ or ( b₂ * (e . b₄) <= b₁ . b₄ & 0 <= b₁ . b₄ ) ) ) ) } is set
x is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() Element of K19(K20(NAT,REAL))
f is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
x . f is V25() real ext-real Element of REAL
sqrt f is V25() real ext-real Element of REAL
f to_power (sqrt f) is V25() real ext-real Element of REAL
x /" (2,1,0) is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() Element of K19(K20(NAT,REAL))
(2,1,0) " is V1() V4( NAT ) Function-like V14( NAT ) V35() V36() V37() set
x (#) ((2,1,0) ") is V1() V4( NAT ) Function-like V14( NAT ) V35() V36() V37() set
e is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() eventually-nonnegative eventually-positive eventually-nonzero Element of K19(K20(NAT,REAL))
Big_Oh e is functional non empty FUNCTION_DOMAIN of NAT , REAL
{ b₁ where b₁ is V1() V4( NAT ) V5( REAL ) Function-like V14( NAT ) quasi_total V35() V36() V37() Element of Funcs (NAT,REAL) : ex b₂ being V25() real ext-real Element of REAL ex b₃ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT st
( not b₂ <= 0 & ( for b₄ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT holds
( not b₃ <= b₄ or ( b₁ . b₄ <= b₂ * (e . b₄) & 0 <= b₁ . b₄ ) ) ) ) } is set
N0 is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() eventually-nonnegative eventually-positive eventually-nonzero Element of K19(K20(NAT,REAL))
Big_Oh N0 is functional non empty FUNCTION_DOMAIN of NAT , REAL
{ b₁ where b₁ is V1() V4( NAT ) V5( REAL ) Function-like V14( NAT ) quasi_total V35() V36() V37() Element of Funcs (NAT,REAL) : ex b₂ being V25() real ext-real Element of REAL ex b₃ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT st
( not b₂ <= 0 & ( for b₄ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT holds
( not b₃ <= b₄ or ( b₁ . b₄ <= b₂ * (N0 . b₄) & 0 <= b₁ . b₄ ) ) ) ) } is set
t is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
t is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
d is V25() real ext-real set
1 / d is V25() real ext-real Element of REAL
d " is V25() real ext-real set
1 * (d ") is V25() real ext-real set
log (2,(1 / d)) is V25() real ext-real Element of REAL
[/(log (2,(1 / d)))\] is V25() real ext-real integer set
2 * [/(log (2,(1 / d)))\] is V25() real ext-real integer Element of REAL
max ((2 * [/(log (2,(1 / d)))\]),2) is V25() real ext-real Element of REAL
max (t,(max ((2 * [/(log (2,(1 / d)))\]),2))) is V25() real ext-real Element of REAL
b is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
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
(x /" (2,1,0)) . n is V25() real ext-real Element of REAL
e . n is V25() real ext-real Element of REAL
N0 . n is V25() real ext-real Element of REAL
(e . n) / (N0 . n) is V25() real ext-real Element of REAL
(N0 . n) " is V25() real ext-real set
(e . n) * ((N0 . n) ") is V25() real ext-real set
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
n / 2 is V25() real ext-real non negative Element of REAL
n * (2 ") is V25() real ext-real non negative set
- (n / 2) is V25() real ext-real non positive Element of REAL
- (log (2,(1 / d))) is V25() real ext-real Element of REAL
2 to_power (- (n / 2)) is V25() real ext-real Element of REAL
2 to_power (- (log (2,(1 / d)))) is V25() real ext-real Element of REAL
2 to_power (log (2,(1 / d))) is V25() real ext-real Element of REAL
1 / (2 to_power (log (2,(1 / d)))) is V25() real ext-real Element of REAL
(2 to_power (log (2,(1 / d)))) " is V25() real ext-real set
1 * ((2 to_power (log (2,(1 / d)))) ") is V25() real ext-real set
1 / (1 / d) is V25() real ext-real Element of REAL
(1 / d) " is V25() real ext-real set
1 * ((1 / d) ") is V25() real ext-real set
1 * 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
(1 * n) + 0 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 to_power ((1 * n) + 0) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 |^ ((1 * n) + 0) is set
2 to_power 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
2 |^ n is set
sqrt n is V25() real ext-real Element of REAL
n to_power (sqrt n) is V25() real ext-real Element of REAL
log (2,n) is V25() real ext-real Element of REAL
(sqrt n) * (log (2,n)) is V25() real ext-real Element of REAL
2 to_power ((sqrt n) * (log (2,n))) is V25() real ext-real Element of REAL
((sqrt n) * (log (2,n))) - n is V25() real ext-real Element of REAL
- n is V25() real ext-real non positive integer set
((sqrt n) * (log (2,n))) + (- n) is V25() real ext-real set
2 to_power (((sqrt n) * (log (2,n))) - n) is V25() real ext-real Element of REAL
n - ((sqrt n) * (log (2,n))) is V25() real ext-real Element of REAL
- ((sqrt n) * (log (2,n))) is V25() real ext-real set
n + (- ((sqrt n) * (log (2,n)))) is V25() real ext-real set
- (n - ((sqrt n) * (log (2,n)))) is V25() real ext-real Element of REAL
2 to_power (- (n - ((sqrt n) * (log (2,n))))) is V25() real ext-real Element of REAL
((x /" (2,1,0)) . n) - 0 is V25() real ext-real Element of REAL
- 0 is functional empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V25() real ext-real non positive non negative integer V49() V50() V51() V52() V53() V54() V55() FinSequence-membered set
((x /" (2,1,0)) . n) + (- 0) is V25() real ext-real set
abs (((x /" (2,1,0)) . n) - 0) is V25() real ext-real Element of REAL
lim (x /" (2,1,0)) is V25() real ext-real Element of REAL
Big_Omega N0 is functional non empty FUNCTION_DOMAIN of NAT , REAL
{ b₁ where b₁ is V1() V4( NAT ) V5( REAL ) Function-like V14( NAT ) quasi_total V35() V36() V37() Element of Funcs (NAT,REAL) : ex b₂ being V25() real ext-real Element of REAL ex b₃ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT st
( not b₂ <= 0 & ( for b₄ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT holds
( not b₃ <= b₄ or ( b₂ * (N0 . b₄) <= b₁ . b₄ & 0 <= b₁ . b₄ ) ) ) ) } is set
(2,1,0) /" (2,1,1) is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() Element of K19(K20(NAT,REAL))
(2,1,1) " is V1() V4( NAT ) Function-like V14( NAT ) V35() V36() V37() set
(2,1,0) (#) ((2,1,1) ") is V1() V4( NAT ) Function-like V14( NAT ) V35() V36() V37() set
e is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() eventually-nonnegative eventually-positive eventually-nonzero Element of K19(K20(NAT,REAL))
Big_Oh e is functional non empty FUNCTION_DOMAIN of NAT , REAL
{ b₁ where b₁ is V1() V4( NAT ) V5( REAL ) Function-like V14( NAT ) quasi_total V35() V36() V37() Element of Funcs (NAT,REAL) : ex b₂ being V25() real ext-real Element of REAL ex b₃ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT st
( not b₂ <= 0 & ( for b₄ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT holds
( not b₃ <= b₄ or ( b₁ . b₄ <= b₂ * (e . b₄) & 0 <= b₁ . b₄ ) ) ) ) } is set
N0 is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() eventually-nonnegative eventually-positive eventually-nonzero Element of K19(K20(NAT,REAL))
Big_Oh N0 is functional non empty FUNCTION_DOMAIN of NAT , REAL
{ b₁ where b₁ is V1() V4( NAT ) V5( REAL ) Function-like V14( NAT ) quasi_total V35() V36() V37() Element of Funcs (NAT,REAL) : ex b₂ being V25() real ext-real Element of REAL ex b₃ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT st
( not b₂ <= 0 & ( for b₄ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT holds
( not b₃ <= b₄ or ( b₁ . b₄ <= b₂ * (N0 . b₄) & 0 <= b₁ . b₄ ) ) ) ) } is set
t is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
N0 . t is V25() real ext-real Element of REAL
1 * t is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(1 * t) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 to_power ((1 * t) + 1) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 |^ ((1 * t) + 1) is set
e . t is V25() real ext-real Element of REAL
(1 * t) + 0 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 to_power ((1 * t) + 0) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 |^ ((1 * t) + 0) is set
((2,1,0) /" (2,1,1)) . t is V25() real ext-real Element of REAL
2 to_power t is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 |^ t is set
(2 to_power t) / (N0 . t) is V25() real ext-real Element of REAL
(N0 . t) " is V25() real ext-real set
(2 to_power t) * ((N0 . t) ") is V25() real ext-real set
t + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
t - (t + 1) is V25() real ext-real integer Element of REAL
- (t + 1) is V25() real ext-real non positive integer set
t + (- (t + 1)) is V25() real ext-real integer set
2 to_power (t - (t + 1)) is V25() real ext-real Element of REAL
0 + (- 1) is V25() real ext-real non positive integer Element of REAL
2 to_power (0 + (- 1)) is V25() real ext-real Element of REAL
2 to_power 1 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 |^ 1 is set
1 / (2 to_power 1) is V25() real ext-real non negative Element of REAL
(2 to_power 1) " is V25() real ext-real non negative set
1 * ((2 to_power 1) ") is V25() real ext-real non negative set
2 " is non empty V25() real ext-real positive non negative Element of REAL
t is V25() real ext-real set
d is 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
N1 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
((2,1,0) /" (2,1,1)) . N1 is V25() real ext-real Element of REAL
(((2,1,0) /" (2,1,1)) . N1) - (2 ") is V25() real ext-real Element of REAL
- (2 ") is V25() real ext-real non positive set
(((2,1,0) /" (2,1,1)) . N1) + (- (2 ")) is V25() real ext-real set
abs ((((2,1,0) /" (2,1,1)) . N1) - (2 ")) is V25() real ext-real Element of REAL
(2 ") - (2 ") is V25() real ext-real Element of REAL
(2 ") + (- (2 ")) is V25() real ext-real set
abs ((2 ") - (2 ")) is V25() real ext-real Element of REAL
lim ((2,1,0) /" (2,1,1)) is V25() real ext-real Element of REAL
x is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() eventually-nonnegative eventually-positive eventually-nonzero Element of K19(K20(NAT,REAL))
Big_Oh x is functional non empty FUNCTION_DOMAIN of NAT , REAL
{ b₁ where b₁ is V1() V4( NAT ) V5( REAL ) Function-like V14( NAT ) quasi_total V35() V36() V37() Element of Funcs (NAT,REAL) : ex b₂ being V25() real ext-real Element of REAL ex b₃ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT st
( not b₂ <= 0 & ( for b₄ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT holds
( not b₃ <= b₄ or ( b₁ . b₄ <= b₂ * (x . b₄) & 0 <= b₁ . b₄ ) ) ) ) } is set
c is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() eventually-nonnegative eventually-positive eventually-nonzero Element of K19(K20(NAT,REAL))
Big_Oh c is functional non empty FUNCTION_DOMAIN of NAT , REAL
{ b₁ where b₁ is V1() V4( NAT ) V5( REAL ) Function-like V14( NAT ) quasi_total V35() V36() V37() Element of Funcs (NAT,REAL) : ex b₂ being V25() real ext-real Element of REAL ex b₃ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT st
( not b₂ <= 0 & ( for b₄ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT holds
( not b₃ <= b₄ or ( b₁ . b₄ <= b₂ * (c . b₄) & 0 <= b₁ . b₄ ) ) ) ) } is set
x /" c is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() Element of K19(K20(NAT,REAL))
c " is V1() V4( NAT ) Function-like V14( NAT ) V35() V36() V37() set
x (#) (c ") is V1() V4( NAT ) Function-like V14( NAT ) V35() V36() V37() set
e is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(x /" c) . e is V25() real ext-real Element of REAL
- e is V25() real ext-real non positive integer Element of REAL
2 to_power (- e) is V25() real ext-real Element of REAL
x . e is V25() real ext-real Element of REAL
c . e is V25() real ext-real Element of REAL
(x . e) / (c . e) is V25() real ext-real Element of REAL
(c . e) " is V25() real ext-real set
(x . e) * ((c . e) ") is V25() real ext-real set
1 * e is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(1 * e) + 0 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 to_power ((1 * e) + 0) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 |^ ((1 * e) + 0) is set
(2 to_power ((1 * e) + 0)) / (c . e) is V25() real ext-real Element of REAL
(2 to_power ((1 * e) + 0)) * ((c . e) ") is V25() real ext-real set
2 to_power (1 * e) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 |^ (1 * e) is set
2 * e is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(2 * e) + 0 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 to_power ((2 * e) + 0) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 |^ ((2 * e) + 0) is set
(2 to_power (1 * e)) / (2 to_power ((2 * e) + 0)) is V25() real ext-real non negative Element of REAL
(2 to_power ((2 * e) + 0)) " is V25() real ext-real non negative set
(2 to_power (1 * e)) * ((2 to_power ((2 * e) + 0)) ") is V25() real ext-real non negative set
(1 * e) - (2 * e) is V25() real ext-real integer Element of REAL
- (2 * e) is V25() real ext-real non positive integer set
(1 * e) + (- (2 * e)) is V25() real ext-real integer set
2 to_power ((1 * e) - (2 * e)) is V25() real ext-real Element of REAL
e is V25() real ext-real set
1 / e is V25() real ext-real Element of REAL
e " is V25() real ext-real set
1 * (e ") is V25() real ext-real set
log (2,(1 / e)) is V25() real ext-real Element of REAL
[/(log (2,(1 / e)))\] is V25() real ext-real integer set
[/(log (2,(1 / e)))\] + 1 is V25() real ext-real integer Element of REAL
max (1,([/(log (2,(1 / e)))\] + 1)) is V25() real ext-real Element of REAL
2 to_power ([/(log (2,(1 / e)))\] + 1) is V25() real ext-real Element of REAL
2 to_power (log (2,(1 / e))) is V25() real ext-real Element of REAL
t is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
d is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 to_power d is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 |^ d is set
N1 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 to_power N1 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 |^ N1 is set
(2 to_power N1) * e is V25() real ext-real Element of REAL
(1 / e) * e is V25() real ext-real Element of REAL
e * (2 to_power N1) is V25() real ext-real Element of REAL
(2 to_power N1) " is V25() real ext-real non negative Element of REAL
(e * (2 to_power N1)) * ((2 to_power N1) ") is V25() real ext-real Element of REAL
1 * ((2 to_power N1) ") is V25() real ext-real non negative Element of REAL
(2 to_power N1) * ((2 to_power N1) ") is V25() real ext-real non negative Element of REAL
e * ((2 to_power N1) * ((2 to_power N1) ")) is V25() real ext-real Element of REAL
e * 1 is V25() real ext-real Element of REAL
1 / (2 to_power N1) is V25() real ext-real non negative Element of REAL
(2 to_power N1) " is V25() real ext-real non negative set
1 * ((2 to_power N1) ") is V25() real ext-real non negative set
- N1 is V25() real ext-real non positive integer Element of REAL
2 to_power (- N1) is V25() real ext-real Element of REAL
(x /" c) . N1 is V25() real ext-real Element of REAL
((x /" c) . N1) - 0 is V25() real ext-real Element of REAL
- 0 is functional empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V25() real ext-real non positive non negative integer V49() V50() V51() V52() V53() V54() V55() FinSequence-membered set
((x /" c) . N1) + (- 0) is V25() real ext-real set
abs (((x /" c) . N1) - 0) is V25() real ext-real Element of REAL
abs (2 to_power (- N1)) is V25() real ext-real Element of REAL
lim (x /" c) is V25() real ext-real Element of REAL
Big_Omega c is functional non empty FUNCTION_DOMAIN of NAT , REAL
{ b₁ where b₁ is V1() V4( NAT ) V5( REAL ) Function-like V14( NAT ) quasi_total V35() V36() V37() Element of Funcs (NAT,REAL) : ex b₂ being V25() real ext-real Element of REAL ex b₃ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT st
( not b₂ <= 0 & ( for b₄ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT holds
( not b₃ <= b₄ or ( b₂ * (c . b₄) <= b₁ . b₄ & 0 <= b₁ . b₄ ) ) ) ) } is set
Big_Oh (0) is functional non empty FUNCTION_DOMAIN of NAT , REAL
{ b₁ where b₁ is V1() V4( NAT ) V5( REAL ) Function-like V14( NAT ) quasi_total V35() V36() V37() Element of Funcs (NAT,REAL) : ex b₂ being V25() real ext-real Element of REAL ex b₃ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT st
( not b₂ <= 0 & ( for b₄ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT holds
( not b₃ <= b₄ or ( b₁ . b₄ <= b₂ * ((0) . b₄) & 0 <= b₁ . b₄ ) ) ) ) } is set
c is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() eventually-nonnegative eventually-positive eventually-nonzero Element of K19(K20(NAT,REAL))
Big_Oh c is functional non empty FUNCTION_DOMAIN of NAT , REAL
{ b₁ where b₁ is V1() V4( NAT ) V5( REAL ) Function-like V14( NAT ) quasi_total V35() V36() V37() Element of Funcs (NAT,REAL) : ex b₂ being V25() real ext-real Element of REAL ex b₃ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT st
( not b₂ <= 0 & ( for b₄ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT holds
( not b₃ <= b₄ or ( b₁ . b₄ <= b₂ * (c . b₄) & 0 <= b₁ . b₄ ) ) ) ) } is set
c /" (0) is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() Element of K19(K20(NAT,REAL))
(0) " is V1() V4( NAT ) Function-like V14( NAT ) V35() V36() V37() set
c (#) ((0) ") is V1() V4( NAT ) Function-like V14( NAT ) V35() V36() V37() set
e is V25() real ext-real set
1 / e is V25() real ext-real Element of REAL
e " is V25() real ext-real set
1 * (e ") is V25() real ext-real set
log (2,(1 / e)) is V25() real ext-real Element of REAL
9 + (log (2,(1 / e))) is V25() real ext-real Element of REAL
[/(9 + (log (2,(1 / e))))\] is V25() real ext-real integer set
max (10,[/(9 + (log (2,(1 / e))))\]) is V25() real ext-real set
t is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
d is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
N1 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
N1 - 9 is V25() real ext-real integer Element of REAL
- 9 is V25() real ext-real non positive integer set
N1 + (- 9) is V25() real ext-real integer set
2 to_power (log (2,(1 / e))) is V25() real ext-real Element of REAL
2 to_power (N1 - 9) is V25() real ext-real Element of REAL
1 / (2 to_power (N1 - 9)) is V25() real ext-real Element of REAL
(2 to_power (N1 - 9)) " is V25() real ext-real set
1 * ((2 to_power (N1 - 9)) ") is V25() real ext-real set
1 / (1 / e) is V25() real ext-real Element of REAL
(1 / e) " is V25() real ext-real set
1 * ((1 / e) ") is V25() real ext-real set
(c /" (0)) . N1 is V25() real ext-real Element of REAL
c . N1 is V25() real ext-real Element of REAL
(0) . N1 is V25() real ext-real Element of REAL
(c . N1) / ((0) . N1) is V25() real ext-real Element of REAL
((0) . N1) " is V25() real ext-real set
(c . N1) * (((0) . N1) ") is V25() real ext-real set
2 * N1 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(2 * N1) + 0 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 to_power ((2 * N1) + 0) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 |^ ((2 * N1) + 0) is set
(2 to_power ((2 * N1) + 0)) / ((0) . N1) is V25() real ext-real Element of REAL
(2 to_power ((2 * N1) + 0)) * (((0) . N1) ") is V25() real ext-real set
N1 + 0 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(N1 + 0) ! is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(2 to_power ((2 * N1) + 0)) / ((N1 + 0) !) is V25() real ext-real non negative Element of REAL
((N1 + 0) !) " is V25() real ext-real non negative set
(2 to_power ((2 * N1) + 0)) * (((N1 + 0) !) ") is V25() real ext-real non negative set
2 to_power (2 * N1) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 |^ (2 * N1) is set
N1 ! is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(2 to_power (2 * N1)) / (N1 !) is V25() real ext-real non negative Element of REAL
(N1 !) " is V25() real ext-real non negative set
(2 to_power (2 * N1)) * ((N1 !) ") is V25() real ext-real non negative set
((c /" (0)) . N1) - 0 is V25() real ext-real Element of REAL
- 0 is functional empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V25() real ext-real non positive non negative integer V49() V50() V51() V52() V53() V54() V55() FinSequence-membered set
((c /" (0)) . N1) + (- 0) is V25() real ext-real set
abs (((c /" (0)) . N1) - 0) is V25() real ext-real Element of REAL
lim (c /" (0)) is V25() real ext-real Element of REAL
Big_Omega (0) is functional non empty FUNCTION_DOMAIN of NAT , REAL
{ b₁ where b₁ is V1() V4( NAT ) V5( REAL ) Function-like V14( NAT ) quasi_total V35() V36() V37() Element of Funcs (NAT,REAL) : ex b₂ being V25() real ext-real Element of REAL ex b₃ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT st
( not b₂ <= 0 & ( for b₄ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT holds
( not b₃ <= b₄ or ( b₂ * ((0) . b₄) <= b₁ . b₄ & 0 <= b₁ . b₄ ) ) ) ) } is set
(0) /" (1) is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() Element of K19(K20(NAT,REAL))
(1) " is V1() V4( NAT ) Function-like V14( NAT ) V35() V36() V37() set
(0) (#) ((1) ") is V1() V4( NAT ) Function-like V14( NAT ) V35() V36() V37() set
e is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
((0) /" (1)) . e is V25() real ext-real Element of REAL
e + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
1 / (e + 1) is V25() real ext-real non negative Element of REAL
(e + 1) " is non empty V25() real ext-real positive non negative set
1 * ((e + 1) ") is V25() real ext-real non negative set
e ! is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(0) . e is V25() real ext-real Element of REAL
(1) . e is V25() real ext-real Element of REAL
((0) . e) / ((1) . e) is V25() real ext-real Element of REAL
((1) . e) " is V25() real ext-real set
((0) . e) * (((1) . e) ") is V25() real ext-real set
e + 0 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(e + 0) ! is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
((e + 0) !) / ((1) . e) is V25() real ext-real Element of REAL
((e + 0) !) * (((1) . e) ") is V25() real ext-real set
(e + 1) ! is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(e !) / ((e + 1) !) is V25() real ext-real non negative Element of REAL
((e + 1) !) " is V25() real ext-real non negative set
(e !) * (((e + 1) !) ") is V25() real ext-real non negative set
(e !) * 1 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(e + 1) * (e !) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
((e !) * 1) / ((e + 1) * (e !)) is V25() real ext-real non negative Element of REAL
((e + 1) * (e !)) " is V25() real ext-real non negative set
((e !) * 1) * (((e + 1) * (e !)) ") is V25() real ext-real non negative set
(e !) / (e !) is V25() real ext-real non negative Element of REAL
(e !) " is V25() real ext-real non negative set
(e !) * ((e !) ") is V25() real ext-real non negative set
(1 / (e + 1)) * ((e !) / (e !)) is V25() real ext-real non negative Element of REAL
(1 / (e + 1)) * 1 is V25() real ext-real non negative Element of REAL
e is V25() real ext-real set
1 / e is V25() real ext-real Element of REAL
e " is V25() real ext-real set
1 * (e ") is V25() real ext-real set
[/(1 / e)\] is V25() real ext-real integer set
max (1,[/(1 / e)\]) is V25() real ext-real set
t is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
d is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
N1 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
N1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
1 / (1 / e) is V25() real ext-real Element of REAL
(1 / e) " is V25() real ext-real set
1 * ((1 / e) ") is V25() real ext-real set
1 / (N1 + 1) is V25() real ext-real non negative Element of REAL
(N1 + 1) " is non empty V25() real ext-real positive non negative set
1 * ((N1 + 1) ") is V25() real ext-real non negative set
((0) /" (1)) . N1 is V25() real ext-real Element of REAL
- 0 is functional empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V25() real ext-real non positive non negative integer V49() V50() V51() V52() V53() V54() V55() FinSequence-membered Element of REAL
- e is V25() real ext-real set
abs (((0) /" (1)) . N1) is V25() real ext-real Element of REAL
(- 1) * (- e) is V25() real ext-real Element of REAL
(- 1) * (((0) /" (1)) . N1) is V25() real ext-real Element of REAL
- (((0) /" (1)) . N1) is V25() real ext-real Element of REAL
(((0) /" (1)) . N1) - 0 is V25() real ext-real Element of REAL
- 0 is functional empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V25() real ext-real non positive non negative integer V49() V50() V51() V52() V53() V54() V55() FinSequence-membered set
(((0) /" (1)) . N1) + (- 0) is V25() real ext-real set
abs ((((0) /" (1)) . N1) - 0) is V25() real ext-real Element of REAL
lim ((0) /" (1)) is V25() real ext-real Element of REAL
x is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() Element of K19(K20(NAT,REAL))
f is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
x . f is V25() real ext-real Element of REAL
f to_power f is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
f |^ f is set
(1) /" x is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() Element of K19(K20(NAT,REAL))
x " is V1() V4( NAT ) Function-like V14( NAT ) V35() V36() V37() set
(1) (#) (x ") is V1() V4( NAT ) Function-like V14( NAT ) V35() V36() V37() set
N0 is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() Element of K19(K20(NAT,REAL))
t is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
N0 . t is V25() real ext-real Element of REAL
t + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
t + 2 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(t + 1) / (t + 2) is V25() real ext-real non negative Element of REAL
(t + 2) " is non empty V25() real ext-real positive non negative set
(t + 1) * ((t + 2) ") is V25() real ext-real non negative set
(t + 1) / t is V25() real ext-real non negative Element of REAL
t " is V25() real ext-real non negative set
(t + 1) * (t ") is V25() real ext-real non negative set
((t + 1) / t) to_power t is V25() real ext-real Element of REAL
((t + 1) / t) |^ t is set
((t + 1) / (t + 2)) * (((t + 1) / t) to_power t) is V25() real ext-real Element of REAL
(t + 1) ! is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
((1) /" x) . t is V25() real ext-real Element of REAL
((1) /" x) . (t + 1) is V25() real ext-real Element of REAL
(((1) /" x) . t) / (((1) /" x) . (t + 1)) is V25() real ext-real Element of REAL
(((1) /" x) . (t + 1)) " is V25() real ext-real set
(((1) /" x) . t) * ((((1) /" x) . (t + 1)) ") is V25() real ext-real set
(1) . t is V25() real ext-real Element of REAL
e is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() eventually-nonnegative eventually-positive eventually-nonzero Element of K19(K20(NAT,REAL))
e . t is V25() real ext-real Element of REAL
((1) . t) / (e . t) is V25() real ext-real Element of REAL
(e . t) " is V25() real ext-real set
((1) . t) * ((e . t) ") is V25() real ext-real set
(((1) . t) / (e . t)) / (((1) /" x) . (t + 1)) is V25() real ext-real Element of REAL
(((1) . t) / (e . t)) * ((((1) /" x) . (t + 1)) ") is V25() real ext-real set
((t + 1) !) / (e . t) is V25() real ext-real Element of REAL
((t + 1) !) * ((e . t) ") is V25() real ext-real set
(((t + 1) !) / (e . t)) / (((1) /" x) . (t + 1)) is V25() real ext-real Element of REAL
(((t + 1) !) / (e . t)) * ((((1) /" x) . (t + 1)) ") is V25() real ext-real set
t to_power t is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
t |^ t is set
((t + 1) !) / (t to_power t) is V25() real ext-real non negative Element of REAL
(t to_power t) " is V25() real ext-real non negative set
((t + 1) !) * ((t to_power t) ") is V25() real ext-real non negative set
(((t + 1) !) / (t to_power t)) / (((1) /" x) . (t + 1)) is V25() real ext-real Element of REAL
(((t + 1) !) / (t to_power t)) * ((((1) /" x) . (t + 1)) ") is V25() real ext-real set
(1) . (t + 1) is V25() real ext-real Element of REAL
e . (t + 1) is V25() real ext-real Element of REAL
((1) . (t + 1)) / (e . (t + 1)) is V25() real ext-real Element of REAL
(e . (t + 1)) " is V25() real ext-real set
((1) . (t + 1)) * ((e . (t + 1)) ") is V25() real ext-real set
(((t + 1) !) / (t to_power t)) / (((1) . (t + 1)) / (e . (t + 1))) is V25() real ext-real Element of REAL
(((1) . (t + 1)) / (e . (t + 1))) " is V25() real ext-real set
(((t + 1) !) / (t to_power t)) * ((((1) . (t + 1)) / (e . (t + 1))) ") is V25() real ext-real set
(t + 1) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
((t + 1) + 1) ! is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(((t + 1) + 1) !) / (e . (t + 1)) is V25() real ext-real Element of REAL
(((t + 1) + 1) !) * ((e . (t + 1)) ") is V25() real ext-real set
(((t + 1) !) / (t to_power t)) / ((((t + 1) + 1) !) / (e . (t + 1))) is V25() real ext-real Element of REAL
((((t + 1) + 1) !) / (e . (t + 1))) " is V25() real ext-real set
(((t + 1) !) / (t to_power t)) * (((((t + 1) + 1) !) / (e . (t + 1))) ") is V25() real ext-real set
(t + 1) to_power (t + 1) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(t + 1) |^ (t + 1) is set
(((t + 1) + 1) !) / ((t + 1) to_power (t + 1)) is V25() real ext-real non negative Element of REAL
((t + 1) to_power (t + 1)) " is V25() real ext-real non negative set
(((t + 1) + 1) !) * (((t + 1) to_power (t + 1)) ") is V25() real ext-real non negative set
(((t + 1) !) / (t to_power t)) / ((((t + 1) + 1) !) / ((t + 1) to_power (t + 1))) is V25() real ext-real non negative Element of REAL
((((t + 1) + 1) !) / ((t + 1) to_power (t + 1))) " is V25() real ext-real non negative set
(((t + 1) !) / (t to_power t)) * (((((t + 1) + 1) !) / ((t + 1) to_power (t + 1))) ") is V25() real ext-real non negative set
((t + 1) !) / (((t + 1) + 1) !) is V25() real ext-real non negative Element of REAL
(((t + 1) + 1) !) " is V25() real ext-real non negative set
((t + 1) !) * ((((t + 1) + 1) !) ") is V25() real ext-real non negative set
((t + 1) to_power (t + 1)) / (t to_power t) is V25() real ext-real non negative Element of REAL
((t + 1) to_power (t + 1)) * ((t to_power t) ") is V25() real ext-real non negative set
(((t + 1) !) / (((t + 1) + 1) !)) * (((t + 1) to_power (t + 1)) / (t to_power t)) is V25() real ext-real non negative Element of REAL
((t + 1) + 1) * ((t + 1) !) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
((t + 1) !) / (((t + 1) + 1) * ((t + 1) !)) is V25() real ext-real non negative Element of REAL
(((t + 1) + 1) * ((t + 1) !)) " is V25() real ext-real non negative set
((t + 1) !) * ((((t + 1) + 1) * ((t + 1) !)) ") is V25() real ext-real non negative set
(((t + 1) !) / (((t + 1) + 1) * ((t + 1) !))) * (((t + 1) to_power (t + 1)) / (t to_power t)) is V25() real ext-real non negative Element of REAL
1 / ((t + 1) + 1) is V25() real ext-real non negative Element of REAL
((t + 1) + 1) " is non empty V25() real ext-real positive non negative set
1 * (((t + 1) + 1) ") is V25() real ext-real non negative set
((t + 1) !) / ((t + 1) !) is V25() real ext-real non negative Element of REAL
((t + 1) !) " is V25() real ext-real non negative set
((t + 1) !) * (((t + 1) !) ") is V25() real ext-real non negative set
(1 / ((t + 1) + 1)) * (((t + 1) !) / ((t + 1) !)) is V25() real ext-real non negative Element of REAL
((1 / ((t + 1) + 1)) * (((t + 1) !) / ((t + 1) !))) * (((t + 1) to_power (t + 1)) / (t to_power t)) is V25() real ext-real non negative Element of REAL
(1 / ((t + 1) + 1)) * 1 is V25() real ext-real non negative Element of REAL
((1 / ((t + 1) + 1)) * 1) * (((t + 1) to_power (t + 1)) / (t to_power t)) is V25() real ext-real non negative Element of REAL
1 / (t + 2) is V25() real ext-real non negative Element of REAL
1 * ((t + 2) ") is V25() real ext-real non negative set
(t + 1) to_power t is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(t + 1) |^ t is set
(t + 1) to_power 1 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(t + 1) |^ 1 is set
((t + 1) to_power t) * ((t + 1) to_power 1) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(((t + 1) to_power t) * ((t + 1) to_power 1)) / (t to_power t) is V25() real ext-real non negative Element of REAL
(((t + 1) to_power t) * ((t + 1) to_power 1)) * ((t to_power t) ") is V25() real ext-real non negative set
(1 / (t + 2)) * ((((t + 1) to_power t) * ((t + 1) to_power 1)) / (t to_power t)) is V25() real ext-real non negative Element of REAL
((t + 1) to_power t) * (t + 1) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(((t + 1) to_power t) * (t + 1)) / (t to_power t) is V25() real ext-real non negative Element of REAL
(((t + 1) to_power t) * (t + 1)) * ((t to_power t) ") is V25() real ext-real non negative set
(1 / (t + 2)) * ((((t + 1) to_power t) * (t + 1)) / (t to_power t)) is V25() real ext-real non negative Element of REAL
(t to_power t) " is V25() real ext-real non negative Element of REAL
(((t + 1) to_power t) * (t + 1)) * ((t to_power t) ") is V25() real ext-real non negative Element of REAL
(1 / (t + 2)) * ((((t + 1) to_power t) * (t + 1)) * ((t to_power t) ")) is V25() real ext-real non negative Element of REAL
((t + 1) to_power t) * ((t to_power t) ") is V25() real ext-real non negative Element of REAL
(((t + 1) to_power t) * ((t to_power t) ")) * (t + 1) is V25() real ext-real non negative Element of REAL
(1 / (t + 2)) * ((((t + 1) to_power t) * ((t to_power t) ")) * (t + 1)) is V25() real ext-real non negative Element of REAL
((t + 1) to_power t) / (t to_power t) is V25() real ext-real non negative Element of REAL
((t + 1) to_power t) * ((t to_power t) ") is V25() real ext-real non negative set
(((t + 1) to_power t) / (t to_power t)) * (t + 1) is V25() real ext-real non negative Element of REAL
(1 / (t + 2)) * ((((t + 1) to_power t) / (t to_power t)) * (t + 1)) is V25() real ext-real non negative Element of REAL
(((t + 1) / t) to_power t) * (t + 1) is V25() real ext-real Element of REAL
(1 / (t + 2)) * ((((t + 1) / t) to_power t) * (t + 1)) is V25() real ext-real Element of REAL
(t + 1) * (1 / (t + 2)) is V25() real ext-real non negative Element of REAL
((t + 1) * (1 / (t + 2))) * (((t + 1) / t) to_power t) is V25() real ext-real Element of REAL
t is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer set
N0 . t is V25() real ext-real Element of REAL
t + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
N0 . (t + 1) is V25() real ext-real Element of REAL
t + 2 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(t + 1) " is non empty V25() real ext-real positive non negative Element of REAL
(t + 2) * ((t + 1) ") is V25() real ext-real non negative Element of REAL
0 * ((t + 1) ") is functional empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V25() real ext-real non positive non negative integer V49() V50() V51() V52() V53() V54() V55() FinSequence-membered Element of REAL
(t + 2) / (t + 1) is V25() real ext-real non negative Element of REAL
(t + 1) " is non empty V25() real ext-real positive non negative set
(t + 2) * ((t + 1) ") is V25() real ext-real non negative set
((t + 2) / (t + 1)) to_power (t + 1) is V25() real ext-real Element of REAL
((t + 2) / (t + 1)) |^ (t + 1) is set
t " is V25() real ext-real non negative set
(t + 1) * (t ") is V25() real ext-real non negative Element of REAL
0 * (t ") is functional empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V25() real ext-real non positive non negative integer V49() V50() V51() V52() V53() V54() V55() FinSequence-membered Element of REAL
(t + 1) / t is V25() real ext-real non negative Element of REAL
(t + 1) * (t ") is V25() real ext-real non negative set
((t + 1) / t) to_power t is V25() real ext-real set
((t + 1) / t) |^ t is set
(((t + 2) / (t + 1)) to_power (t + 1)) " is V25() real ext-real Element of REAL
(((t + 1) / t) to_power t) * ((((t + 2) / (t + 1)) to_power (t + 1)) ") is V25() real ext-real Element of REAL
0 * ((((t + 2) / (t + 1)) to_power (t + 1)) ") is V25() real ext-real Element of REAL
(t + 2) * (t + 2) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
t + 3 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(t + 1) * (t + 3) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
t * t is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer set
2 * t is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 * (2 * t) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(t * t) + (2 * (2 * t)) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 ^2 is V25() real ext-real Element of REAL
2 * 2 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer set
((t * t) + (2 * (2 * t))) + (2 ^2) is V25() real ext-real Element of REAL
4 * t is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(t * t) + (4 * t) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
((t * t) + (4 * t)) + 3 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
1 * ((t + 2) * (t + 2)) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
((t + 1) * (t + 3)) / ((t + 2) * (t + 2)) is V25() real ext-real non negative Element of REAL
((t + 2) * (t + 2)) " is V25() real ext-real non negative set
((t + 1) * (t + 3)) * (((t + 2) * (t + 2)) ") is V25() real ext-real non negative set
0 * (t + 3) is 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
((t + 2) * (t + 2)) " is V25() real ext-real non negative Element of REAL
((t + 1) * (t + 3)) * (((t + 2) * (t + 2)) ") is V25() real ext-real non negative Element of REAL
0 * (((t + 2) * (t + 2)) ") is functional empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V25() real ext-real non positive non negative integer V49() V50() V51() V52() V53() V54() V55() FinSequence-membered Element of REAL
1 * (((t + 2) / (t + 1)) to_power (t + 1)) is V25() real ext-real Element of REAL
(((t + 1) / t) to_power t) / (((t + 2) / (t + 1)) to_power (t + 1)) is V25() real ext-real Element of REAL
(((t + 2) / (t + 1)) to_power (t + 1)) " is V25() real ext-real set
(((t + 1) / t) to_power t) * ((((t + 2) / (t + 1)) to_power (t + 1)) ") is V25() real ext-real set
1 * 1 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(((t + 1) * (t + 3)) / ((t + 2) * (t + 2))) * ((((t + 1) / t) to_power t) / (((t + 2) / (t + 1)) to_power (t + 1))) is V25() real ext-real Element of REAL
(t + 3) " is non empty V25() real ext-real positive non negative Element of REAL
(t + 2) * ((t + 3) ") is V25() real ext-real non negative Element of REAL
0 * ((t + 3) ") is functional empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V25() real ext-real non positive non negative integer V49() V50() V51() V52() V53() V54() V55() FinSequence-membered Element of REAL
(t + 2) / (t + 3) is V25() real ext-real non negative Element of REAL
(t + 3) " is non empty V25() real ext-real positive non negative set
(t + 2) * ((t + 3) ") is V25() real ext-real non negative set
((t + 2) / (t + 3)) * (((t + 2) / (t + 1)) to_power (t + 1)) is V25() real ext-real Element of REAL
((t + 2) / (t + 3)) * 0 is functional empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V25() real ext-real non positive non negative integer V49() V50() V51() V52() V53() V54() V55() FinSequence-membered Element of REAL
(t + 1) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(t + 1) + 2 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
((t + 1) + 1) / ((t + 1) + 2) is V25() real ext-real non negative Element of REAL
((t + 1) + 2) " is non empty V25() real ext-real positive non negative set
((t + 1) + 1) * (((t + 1) + 2) ") is V25() real ext-real non negative set
1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
t + (1 + 1) is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(t + (1 + 1)) / (t + 1) is V25() real ext-real non negative Element of REAL
(t + (1 + 1)) * ((t + 1) ") is V25() real ext-real non negative set
((t + (1 + 1)) / (t + 1)) to_power (t + 1) is V25() real ext-real Element of REAL
((t + (1 + 1)) / (t + 1)) |^ (t + 1) is set
(((t + 1) + 1) / ((t + 1) + 2)) * (((t + (1 + 1)) / (t + 1)) to_power (t + 1)) is V25() real ext-real Element of REAL
(N0 . t) / (N0 . (t + 1)) is V25() real ext-real Element of REAL
(N0 . (t + 1)) " is V25() real ext-real set
(N0 . t) * ((N0 . (t + 1)) ") is V25() real ext-real set
(t + 1) / (t + 2) is V25() real ext-real non negative Element of REAL
(t + 2) " is non empty V25() real ext-real positive non negative set
(t + 1) * ((t + 2) ") is V25() real ext-real non negative set
((t + 1) / (t + 2)) * (((t + 1) / t) to_power t) is V25() real ext-real Element of REAL
(((t + 1) / (t + 2)) * (((t + 1) / t) to_power t)) / (((t + 2) / (t + 3)) * (((t + 2) / (t + 1)) to_power (t + 1))) is V25() real ext-real Element of REAL
(((t + 2) / (t + 3)) * (((t + 2) / (t + 1)) to_power (t + 1))) " is V25() real ext-real set
(((t + 1) / (t + 2)) * (((t + 1) / t) to_power t)) * ((((t + 2) / (t + 3)) * (((t + 2) / (t + 1)) to_power (t + 1))) ") is V25() real ext-real set
((t + 1) / (t + 2)) / ((t + 2) / (t + 3)) is V25() real ext-real non negative Element of REAL
((t + 2) / (t + 3)) " is V25() real ext-real non negative set
((t + 1) / (t + 2)) * (((t + 2) / (t + 3)) ") is V25() real ext-real non negative set
(((t + 1) / (t + 2)) / ((t + 2) / (t + 3))) * ((((t + 1) / t) to_power t) / (((t + 2) / (t + 1)) to_power (t + 1))) is V25() real ext-real Element of REAL
1 * (N0 . (t + 1)) is V25() real ext-real Element of REAL
((N0 . t) / (N0 . (t + 1))) * (N0 . (t + 1)) is V25() real ext-real Element of REAL
e is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() eventually-nonnegative eventually-positive eventually-nonzero Element of K19(K20(NAT,REAL))
Big_Oh e is functional non empty FUNCTION_DOMAIN of NAT , REAL
{ b₁ where b₁ is V1() V4( NAT ) V5( REAL ) Function-like V14( NAT ) quasi_total V35() V36() V37() Element of Funcs (NAT,REAL) : ex b₂ being V25() real ext-real Element of REAL ex b₃ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT st
( not b₂ <= 0 & ( for b₄ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT holds
( not b₃ <= b₄ or ( b₁ . b₄ <= b₂ * (e . b₄) & 0 <= b₁ . b₄ ) ) ) ) } is set
t is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
((1) /" x) . t is V25() real ext-real Element of REAL
t + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(t + 1) ! is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
t to_power t is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
t |^ t is set
1 / (t to_power t) is V25() real ext-real non negative Element of REAL
(t to_power t) " is V25() real ext-real non negative set
1 * ((t to_power t) ") is V25() real ext-real non negative set
((t + 1) !) * (1 / (t to_power t)) is V25() real ext-real non negative Element of REAL
((t + 1) !) * 0 is 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
(1) . t is V25() real ext-real Element of REAL
e . t is V25() real ext-real Element of REAL
((1) . t) / (e . t) is V25() real ext-real Element of REAL
(e . t) " is V25() real ext-real set
((1) . t) * ((e . t) ") is V25() real ext-real set
((t + 1) !) / (e . t) is V25() real ext-real Element of REAL
((t + 1) !) * ((e . t) ") is V25() real ext-real set
((t + 1) !) / (t to_power t) is V25() real ext-real non negative Element of REAL
((t + 1) !) * ((t to_power t) ") is V25() real ext-real non negative set
N0 . 4 is V25() real ext-real Element of REAL
4 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
4 + 2 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(4 + 1) / (4 + 2) is V25() real ext-real non negative Element of REAL
(4 + 2) " is non empty V25() real ext-real positive non negative set
(4 + 1) * ((4 + 2) ") is V25() real ext-real non negative set
(4 + 1) / 4 is V25() real ext-real non negative Element of REAL
4 " is non empty V25() real ext-real positive non negative set
(4 + 1) * (4 ") is V25() real ext-real non negative set
((4 + 1) / 4) to_power 4 is V25() real ext-real Element of REAL
((4 + 1) / 4) |^ 4 is set
((4 + 1) / (4 + 2)) * (((4 + 1) / 4) to_power 4) is V25() real ext-real Element of REAL
5 / 6 is V25() real ext-real non negative Element of REAL
6 " is non empty V25() real ext-real positive non negative set
5 * (6 ") is V25() real ext-real non negative set
5 to_power 4 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
5 |^ 4 is set
(5 to_power 4) / 256 is V25() real ext-real non negative Element of REAL
256 " is non empty V25() real ext-real positive non negative set
(5 to_power 4) * (256 ") is V25() real ext-real non negative set
(5 / 6) * ((5 to_power 4) / 256) is V25() real ext-real non negative Element of REAL
5 * (5 to_power 4) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
6 * 256 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(5 * (5 to_power 4)) / (6 * 256) is V25() real ext-real non negative Element of REAL
(6 * 256) " is V25() real ext-real non negative set
(5 * (5 to_power 4)) * ((6 * 256) ") is V25() real ext-real non negative set
5 to_power 1 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
5 |^ 1 is set
(5 to_power 1) * (5 to_power 4) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
1536 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
((5 to_power 1) * (5 to_power 4)) / 1536 is V25() real ext-real non negative Element of REAL
1536 " is non empty V25() real ext-real positive non negative set
((5 to_power 1) * (5 to_power 4)) * (1536 ") is V25() real ext-real non negative set
5 to_power (4 + 1) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
5 |^ (4 + 1) is set
(5 to_power (4 + 1)) / 1536 is V25() real ext-real non negative Element of REAL
(5 to_power (4 + 1)) * (1536 ") is V25() real ext-real non negative set
3125 / 1536 is V25() real ext-real non negative Element of REAL
3125 * (1536 ") is V25() real ext-real non negative set
4 - 1 is V25() real ext-real integer Element of REAL
- 1 is V25() real ext-real non positive integer set
4 + (- 1) is V25() real ext-real integer set
t is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer set
t - 2 is V25() real ext-real integer Element of REAL
- 2 is V25() real ext-real non positive integer set
t + (- 2) is V25() real ext-real integer set
1 / (t - 2) is V25() real ext-real Element of REAL
(t - 2) " is V25() real ext-real set
1 * ((t - 2) ") is V25() real ext-real set
((1) /" x) . t is V25() real ext-real Element of REAL
t + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(t + 1) - 2 is V25() real ext-real integer Element of REAL
(t + 1) + (- 2) is V25() real ext-real integer set
1 / ((t + 1) - 2) is V25() real ext-real Element of REAL
((t + 1) - 2) " is V25() real ext-real set
1 * (((t + 1) - 2) ") is V25() real ext-real set
((1) /" x) . (t + 1) is V25() real ext-real Element of REAL
N0 . t is V25() real ext-real Element of REAL
(((1) /" x) . t) / (((1) /" x) . (t + 1)) is V25() real ext-real Element of REAL
(((1) /" x) . (t + 1)) " is V25() real ext-real set
(((1) /" x) . t) * ((((1) /" x) . (t + 1)) ") is V25() real ext-real set
((((1) /" x) . t) / (((1) /" x) . (t + 1))) * (((1) /" x) . (t + 1)) is V25() real ext-real Element of REAL
2 * (((1) /" x) . (t + 1)) is V25() real ext-real Element of REAL
(((1) /" x) . t) / 2 is V25() real ext-real Element of REAL
(((1) /" x) . t) * (2 ") is V25() real ext-real set
t - 1 is V25() real ext-real integer Element of REAL
t + (- 1) is V25() real ext-real integer set
2 * (t - 2) is V25() real ext-real integer Element of REAL
1 / (2 * (t - 2)) is V25() real ext-real Element of REAL
(2 * (t - 2)) " is V25() real ext-real set
1 * ((2 * (t - 2)) ") is V25() real ext-real set
1 / (t - 1) is V25() real ext-real Element of REAL
(t - 1) " is V25() real ext-real set
1 * ((t - 1) ") is V25() real ext-real set
(1 / (t - 2)) * (1 / 2) is V25() real ext-real Element of REAL
(((1) /" x) . t) * (1 / 2) is V25() real ext-real Element of REAL
((1) /" x) . 4 is V25() real ext-real Element of REAL
(1) . 4 is V25() real ext-real Element of REAL
e . 4 is V25() real ext-real Element of REAL
((1) . 4) / (e . 4) is V25() real ext-real Element of REAL
(e . 4) " is V25() real ext-real set
((1) . 4) * ((e . 4) ") is V25() real ext-real set
(4 + 1) ! is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
((4 + 1) !) / (e . 4) is V25() real ext-real Element of REAL
((4 + 1) !) * ((e . 4) ") is V25() real ext-real set
120 / 256 is V25() real ext-real non negative Element of REAL
120 * (256 ") is V25() real ext-real non negative set
4 - 2 is V25() real ext-real integer Element of REAL
- 2 is V25() real ext-real non positive integer set
4 + (- 2) is V25() real ext-real integer set
1 / (4 - 2) is V25() real ext-real Element of REAL
(4 - 2) " is V25() real ext-real set
1 * ((4 - 2) ") is V25() real ext-real set
t is V25() real ext-real set
1 / t is V25() real ext-real Element of REAL
t " is V25() real ext-real set
1 * (t ") is V25() real ext-real set
(1 / t) + 4 is V25() real ext-real Element of REAL
[/((1 / t) + 4)\] is V25() real ext-real integer set
4 + (1 / t) is V25() real ext-real Element of REAL
N1 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
b is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
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
((1) /" x) . N is V25() real ext-real Element of REAL
(1 / t) + 2 is V25() real ext-real Element of REAL
((1 / t) + 2) + 2 is V25() real ext-real Element of REAL
N - 2 is V25() real ext-real integer Element of REAL
N + (- 2) is V25() real ext-real integer set
1 / (1 / t) is V25() real ext-real Element of REAL
(1 / t) " is V25() real ext-real set
1 * ((1 / t) ") is V25() real ext-real set
1 / (N - 2) is V25() real ext-real Element of REAL
(N - 2) " is V25() real ext-real set
1 * ((N - 2) ") is V25() real ext-real set
(((1) /" x) . N) - 0 is V25() real ext-real Element of REAL
- 0 is functional empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V25() real ext-real non positive non negative integer V49() V50() V51() V52() V53() V54() V55() FinSequence-membered set
(((1) /" x) . N) + (- 0) is V25() real ext-real set
abs ((((1) /" x) . N) - 0) is V25() real ext-real Element of REAL
lim ((1) /" x) is V25() real ext-real Element of REAL
Big_Omega e is functional non empty FUNCTION_DOMAIN of NAT , REAL
{ b₁ where b₁ is V1() V4( NAT ) V5( REAL ) Function-like V14( NAT ) quasi_total V35() V36() V37() Element of Funcs (NAT,REAL) : ex b₂ being V25() real ext-real Element of REAL ex b₃ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT st
( not b₂ <= 0 & ( for b₄ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT holds
( not b₃ <= b₄ or ( b₂ * (e . b₄) <= b₁ . b₄ & 0 <= b₁ . b₄ ) ) ) ) } is set
c is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
x is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
x + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(x + 1) choose c is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
((x + 1) choose c) / (x + 1) is V25() real ext-real non negative Element of REAL
(x + 1) " is non empty V25() real ext-real positive non negative set
((x + 1) choose c) * ((x + 1) ") is V25() real ext-real non negative set
x choose c is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
x - c is V25() real ext-real integer Element of REAL
- c is V25() real ext-real non positive integer set
x + (- c) is V25() real ext-real integer set
e is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
e + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(x + 1) - c is V25() real ext-real integer Element of REAL
(x + 1) + (- c) is V25() real ext-real integer set
0 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
1 / (e + 1) is V25() real ext-real non negative Element of REAL
(e + 1) " is non empty V25() real ext-real positive non negative set
1 * ((e + 1) ") is V25() real ext-real non negative set
1 / 1 is V25() real ext-real non negative Element of REAL
1 " is non empty V25() real ext-real positive non negative set
1 * (1 ") is V25() real ext-real non negative set
(x choose c) * (1 / (e + 1)) is V25() real ext-real non negative Element of REAL
(x choose c) * (1 / 1) is V25() real ext-real non negative Element of REAL
x + 0 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(x + 1) ! is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
c ! is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(e + 1) ! is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(c !) * ((e + 1) !) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
((x + 1) !) / ((c !) * ((e + 1) !)) is V25() real ext-real non negative Element of REAL
((c !) * ((e + 1) !)) " is V25() real ext-real non negative set
((x + 1) !) * (((c !) * ((e + 1) !)) ") is V25() real ext-real non negative set
(((x + 1) !) / ((c !) * ((e + 1) !))) / (x + 1) is V25() real ext-real non negative Element of REAL
(((x + 1) !) / ((c !) * ((e + 1) !))) * ((x + 1) ") is V25() real ext-real non negative set
x ! is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(x + 1) * (x !) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
((x + 1) * (x !)) / ((c !) * ((e + 1) !)) is V25() real ext-real non negative Element of REAL
((x + 1) * (x !)) * (((c !) * ((e + 1) !)) ") is V25() real ext-real non negative set
(x + 1) * 1 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(((x + 1) * (x !)) / ((c !) * ((e + 1) !))) / ((x + 1) * 1) is V25() real ext-real non negative Element of REAL
((x + 1) * 1) " is V25() real ext-real non negative set
(((x + 1) * (x !)) / ((c !) * ((e + 1) !))) * (((x + 1) * 1) ") is V25() real ext-real non negative set
((c !) * ((e + 1) !)) " is V25() real ext-real non negative Element of REAL
((x + 1) * (x !)) * (((c !) * ((e + 1) !)) ") is V25() real ext-real non negative Element of REAL
(((x + 1) * (x !)) * (((c !) * ((e + 1) !)) ")) / ((x + 1) * 1) is V25() real ext-real non negative Element of REAL
(((x + 1) * (x !)) * (((c !) * ((e + 1) !)) ")) * (((x + 1) * 1) ") is V25() real ext-real non negative set
(x !) * (((c !) * ((e + 1) !)) ") is V25() real ext-real non negative Element of REAL
(x + 1) * ((x !) * (((c !) * ((e + 1) !)) ")) is V25() real ext-real non negative Element of REAL
((x + 1) * ((x !) * (((c !) * ((e + 1) !)) "))) / ((x + 1) * 1) is V25() real ext-real non negative Element of REAL
((x + 1) * ((x !) * (((c !) * ((e + 1) !)) "))) * (((x + 1) * 1) ") is V25() real ext-real non negative set
(x !) / ((c !) * ((e + 1) !)) is V25() real ext-real non negative Element of REAL
(x !) * (((c !) * ((e + 1) !)) ") is V25() real ext-real non negative set
(x + 1) * ((x !) / ((c !) * ((e + 1) !))) is V25() real ext-real non negative Element of REAL
((x + 1) * ((x !) / ((c !) * ((e + 1) !)))) / ((x + 1) * 1) is V25() real ext-real non negative Element of REAL
((x + 1) * ((x !) / ((c !) * ((e + 1) !)))) * (((x + 1) * 1) ") is V25() real ext-real non negative set
(x + 1) / (x + 1) is V25() real ext-real non negative Element of REAL
(x + 1) * ((x + 1) ") is V25() real ext-real non negative set
((x !) / ((c !) * ((e + 1) !))) / 1 is V25() real ext-real non negative Element of REAL
((x !) / ((c !) * ((e + 1) !))) * (1 ") is V25() real ext-real non negative set
((x + 1) / (x + 1)) * (((x !) / ((c !) * ((e + 1) !))) / 1) is V25() real ext-real non negative Element of REAL
1 * (((x !) / ((c !) * ((e + 1) !))) / 1) is V25() real ext-real non negative Element of REAL
e ! is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(e !) * (e + 1) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(c !) * ((e !) * (e + 1)) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(x !) / ((c !) * ((e !) * (e + 1))) is V25() real ext-real non negative Element of REAL
((c !) * ((e !) * (e + 1))) " is V25() real ext-real non negative set
(x !) * (((c !) * ((e !) * (e + 1))) ") is V25() real ext-real non negative set
(x !) * 1 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(c !) * (e !) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
((c !) * (e !)) * (e + 1) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
((x !) * 1) / (((c !) * (e !)) * (e + 1)) is V25() real ext-real non negative Element of REAL
(((c !) * (e !)) * (e + 1)) " is V25() real ext-real non negative set
((x !) * 1) * ((((c !) * (e !)) * (e + 1)) ") is V25() real ext-real non negative set
(x !) / ((c !) * (e !)) is V25() real ext-real non negative Element of REAL
((c !) * (e !)) " is V25() real ext-real non negative set
(x !) * (((c !) * (e !)) ") is V25() real ext-real non negative set
((x !) / ((c !) * (e !))) * (1 / (e + 1)) is V25() real ext-real non negative Element of REAL
c is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer set
c + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
f is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() Element of K19(K20(NAT,REAL))
f . (c + 1) is V25() real ext-real Element of REAL
e is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(e) is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() eventually-nonnegative eventually-positive eventually-nonzero Element of K19(K20(NAT,REAL))
e + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(c + 1) to_power (e + 1) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(c + 1) |^ (e + 1) is set
((c + 1) to_power (e + 1)) / (e + 1) is V25() real ext-real non negative Element of REAL
(e + 1) " is non empty V25() real ext-real positive non negative set
((c + 1) to_power (e + 1)) * ((e + 1) ") is V25() real ext-real non negative set
x is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(x,1) In_Power (e + 1) is V1() V4( NAT ) V5( REAL ) Function-like V35() V36() V37() V56() FinSequence-like FinSubsequence-like FinSequence of REAL
len ((x,1) In_Power (e + 1)) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(e + 1) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
e + 2 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(e + 2) -tuples_on REAL is functional non empty FinSequence-membered FinSequenceSet of REAL
(e + 1) " is non empty V25() real ext-real positive non negative Element of REAL
((e + 1) ") * ((x,1) In_Power (e + 1)) is V1() V4( NAT ) V5( REAL ) Function-like V35() V36() V37() V56() FinSequence-like FinSubsequence-like FinSequence of REAL
len (((e + 1) ") * ((x,1) In_Power (e + 1))) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
x to_power (e + 1) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
x |^ (e + 1) is set
(x to_power (e + 1)) / (e + 1) is V25() real ext-real non negative Element of REAL
(x to_power (e + 1)) * ((e + 1) ") is V25() real ext-real non negative set
<*((x to_power (e + 1)) / (e + 1))*> is V1() V4( NAT ) V5( REAL ) Function-like non empty V35() V36() V37() V56() V67(1) FinSequence-like FinSubsequence-like FinSequence of REAL
(x,1) In_Power e is V1() V4( NAT ) V5( REAL ) Function-like V35() V36() V37() V56() FinSequence-like FinSubsequence-like FinSequence of REAL
<*((x to_power (e + 1)) / (e + 1))*> ^ ((x,1) In_Power e) is V1() V4( NAT ) V5( REAL ) Function-like non empty V35() V36() V37() V56() FinSequence-like FinSubsequence-like FinSequence of REAL
len <*((x to_power (e + 1)) / (e + 1))*> is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
f . x is V25() real ext-real Element of REAL
Sum ((e),x) is V25() real ext-real Element of REAL
Partial_Sums (e) is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() Element of K19(K20(NAT,REAL))
(Partial_Sums (e)) . x is V25() real ext-real Element of REAL
x + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(e) . (x + 1) is V25() real ext-real Element of REAL
(x + 1) to_power e is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(x + 1) |^ e is set
Sum ((x,1) In_Power e) is V25() real ext-real Element of REAL
((x to_power (e + 1)) / (e + 1)) + ((e) . (x + 1)) is V25() real ext-real Element of REAL
Sum (<*((x to_power (e + 1)) / (e + 1))*> ^ ((x,1) In_Power e)) is V25() real ext-real Element of REAL
len ((x,1) In_Power e) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
len (<*((x to_power (e + 1)) / (e + 1))*> ^ ((x,1) In_Power e)) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
Seg (e + 2) is non empty V49() V50() V51() V52() V53() V54() V56() V67(e + 2) Element of K19(NAT)
N1 is V1() V4( NAT ) V5( REAL ) Function-like V35() V36() V37() V56() V67(e + 2) FinSequence-like FinSubsequence-like Element of (e + 2) -tuples_on REAL
n is V1() V4( NAT ) V5( REAL ) Function-like V35() V36() V37() V56() V67(e + 2) FinSequence-like FinSubsequence-like Element of (e + 2) -tuples_on REAL
N0 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer set
N1 . N0 is V25() real ext-real Element of REAL
n . N0 is V25() real ext-real Element of REAL
x |^ (e + 1) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
t is V1() V4( NAT ) V5( REAL ) Function-like V35() V36() V37() V56() V67(e + 2) FinSequence-like FinSubsequence-like Element of (e + 2) -tuples_on REAL
t . 1 is V25() real ext-real Element of REAL
((e + 1) ") * (x |^ (e + 1)) is V25() real ext-real non negative Element of REAL
N0 - 1 is V25() real ext-real integer Element of REAL
- 1 is V25() real ext-real non positive integer set
N0 + (- 1) is V25() real ext-real integer set
(N0 - 1) - 1 is V25() real ext-real integer Element of REAL
(N0 - 1) + (- 1) is V25() real ext-real integer set
1 - 1 is V25() real ext-real integer Element of REAL
1 + (- 1) is V25() real ext-real integer set
e - ((N0 - 1) - 1) is V25() real ext-real integer Element of REAL
- ((N0 - 1) - 1) is V25() real ext-real integer set
e + (- ((N0 - 1) - 1)) is V25() real ext-real integer set
0 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
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
(e + 1) - N is V25() real ext-real integer Element of REAL
- N is V25() real ext-real non positive integer set
(e + 1) + (- N) is V25() real ext-real integer set
dom ((x,1) In_Power e) is V49() V50() V51() V52() V53() V54() Element of K19(NAT)
Seg (e + 1) is non empty V49() V50() V51() V52() V53() V54() V56() V67(e + 1) Element of K19(NAT)
(e + 2) - 1 is V25() real ext-real integer Element of REAL
(e + 2) + (- 1) is V25() real ext-real integer set
m is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
N0 - 2 is V25() real ext-real integer Element of REAL
- 2 is V25() real ext-real non positive integer set
N0 + (- 2) is V25() real ext-real integer set
m + 0 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
l is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
N + 0 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
dom ((x,1) In_Power (e + 1)) is V49() V50() V51() V52() V53() V54() Element of K19(NAT)
t is V1() V4( NAT ) V5( REAL ) Function-like V35() V36() V37() V56() V67(e + 2) FinSequence-like FinSubsequence-like Element of (e + 2) -tuples_on REAL
t . N0 is V25() real ext-real Element of REAL
(e + 1) choose 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
i3 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
x |^ i3 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
((e + 1) choose N) * (x |^ i3) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
1 |^ 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
(((e + 1) choose N) * (x |^ i3)) * (1 |^ 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
((e + 1) ") * ((((e + 1) choose N) * (x |^ i3)) * (1 |^ N)) is V25() real ext-real non negative Element of REAL
(e + 1) choose l is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
x |^ l is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
((e + 1) choose l) * (x |^ l) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(((e + 1) choose l) * (x |^ l)) * (1 |^ 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
((e + 1) ") * ((((e + 1) choose l) * (x |^ l)) * (1 |^ N)) is V25() real ext-real non negative Element of REAL
(((e + 1) choose l) * (x |^ l)) * 1 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
((e + 1) ") * ((((e + 1) choose l) * (x |^ l)) * 1) is V25() real ext-real non negative Element of REAL
((e + 1) ") * ((e + 1) choose l) is V25() real ext-real non negative Element of REAL
x to_power l is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
x |^ l is set
(((e + 1) ") * ((e + 1) choose l)) * (x to_power l) is V25() real ext-real non negative Element of REAL
e - m is V25() real ext-real integer Element of REAL
- m is V25() real ext-real non positive integer set
e + (- m) is V25() real ext-real integer set
e - 0 is V25() real ext-real non negative integer Element of REAL
- 0 is functional empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V25() real ext-real non positive non negative integer V49() V50() V51() V52() V53() V54() V55() FinSequence-membered set
e + (- 0) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer set
((e + 1) choose l) / (e + 1) is V25() real ext-real non negative Element of REAL
((e + 1) choose l) * ((e + 1) ") is V25() real ext-real non negative set
e choose l is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
((x,1) In_Power e) . N is V25() real ext-real Element of REAL
e choose m is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(e choose m) * (x |^ l) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
1 |^ m is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
((e choose m) * (x |^ l)) * (1 |^ m) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(e choose l) * (x |^ l) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
((e choose l) * (x |^ l)) * (1 |^ m) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
((e choose l) * (x |^ l)) * 1 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(e choose l) * (x to_power l) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(x + 1) to_power (e + 1) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(x + 1) |^ (e + 1) is set
((x + 1) to_power (e + 1)) / (e + 1) is V25() real ext-real non negative Element of REAL
((x + 1) to_power (e + 1)) * ((e + 1) ") is V25() real ext-real non negative set
(x + 1) |^ (e + 1) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
((x + 1) |^ (e + 1)) * ((e + 1) ") is V25() real ext-real non negative Element of REAL
Sum ((x,1) In_Power (e + 1)) is V25() real ext-real Element of REAL
(Sum ((x,1) In_Power (e + 1))) * ((e + 1) ") is V25() real ext-real Element of REAL
Sum (((e + 1) ") * ((x,1) In_Power (e + 1))) is V25() real ext-real Element of REAL
Sum n is V25() real ext-real Element of REAL
f . (x + 1) is V25() real ext-real Element of REAL
Sum ((e),(x + 1)) is V25() real ext-real Element of REAL
(Partial_Sums (e)) . (x + 1) is V25() real ext-real Element of REAL
((Partial_Sums (e)) . x) + ((e) . (x + 1)) is V25() real ext-real Element of REAL
c is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() Element of K19(K20(NAT,REAL))
c . 1 is V25() real ext-real Element of REAL
x is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(x) is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() eventually-nonnegative eventually-positive eventually-nonzero Element of K19(K20(NAT,REAL))
x + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
1 to_power (x + 1) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
1 |^ (x + 1) is set
(1 to_power (x + 1)) / (x + 1) is V25() real ext-real non negative Element of REAL
(x + 1) " is non empty V25() real ext-real positive non negative set
(1 to_power (x + 1)) * ((x + 1) ") is V25() real ext-real non negative set
1 / (x + 1) is V25() real ext-real non negative Element of REAL
1 * ((x + 1) ") is V25() real ext-real non negative set
0 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
Sum ((x),1) is V25() real ext-real Element of REAL
Partial_Sums (x) is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() Element of K19(K20(NAT,REAL))
(Partial_Sums (x)) . (0 + 1) is V25() real ext-real Element of REAL
(Partial_Sums (x)) . 0 is V25() real ext-real Element of REAL
(x) . 1 is V25() real ext-real Element of REAL
((Partial_Sums (x)) . 0) + ((x) . 1) is V25() real ext-real Element of REAL
(x) . 0 is V25() real ext-real Element of REAL
((x) . 1) + ((x) . 0) is V25() real ext-real Element of REAL
1 to_power x is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
1 |^ x is set
(1 to_power x) + ((x) . 0) is V25() real ext-real Element of REAL
1 + ((x) . 0) is V25() real ext-real Element of REAL
1 + 0 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
1 / 1 is V25() real ext-real non negative Element of REAL
1 " is non empty V25() real ext-real positive non negative set
1 * (1 ") is V25() real ext-real non negative set
c is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
x is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() Element of K19(K20(NAT,REAL))
f is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(f) is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() eventually-nonnegative eventually-positive eventually-nonzero Element of K19(K20(NAT,REAL))
f + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
c to_power (f + 1) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
c |^ (f + 1) is set
(c to_power (f + 1)) / (f + 1) is V25() real ext-real non negative Element of REAL
(f + 1) " is non empty V25() real ext-real positive non negative set
(c to_power (f + 1)) * ((f + 1) ") is V25() real ext-real non negative set
x . c is V25() real ext-real Element of REAL
x is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() Element of K19(K20(NAT,REAL))
f is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
x . f is V25() real ext-real Element of REAL
Sum ((),f) is V25() real ext-real Element of REAL
f + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
x . (f + 1) is V25() real ext-real Element of REAL
Sum ((),(f + 1)) is V25() real ext-real Element of REAL
f ! is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(f + 1) ! is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
log (2,((f + 1) !)) is V25() real ext-real Element of REAL
(f + 1) * (f !) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
log (2,((f + 1) * (f !))) is V25() real ext-real Element of REAL
log (2,(f + 1)) is V25() real ext-real Element of REAL
log (2,(f !)) is V25() real ext-real Element of REAL
(log (2,(f + 1))) + (log (2,(f !))) is V25() real ext-real Element of REAL
(log (2,(f + 1))) + (Sum ((),f)) is V25() real ext-real Element of REAL
() . (f + 1) is V25() real ext-real Element of REAL
(() . (f + 1)) + (Sum ((),f)) is V25() real ext-real Element of REAL
Partial_Sums () is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() Element of K19(K20(NAT,REAL))
(Partial_Sums ()) . f is V25() real ext-real Element of REAL
(() . (f + 1)) + ((Partial_Sums ()) . f) is V25() real ext-real Element of REAL
(Partial_Sums ()) . (f + 1) is V25() real ext-real Element of REAL
Sum ((),0) is V25() real ext-real Element of REAL
Partial_Sums () is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() Element of K19(K20(NAT,REAL))
(Partial_Sums ()) . 0 is V25() real ext-real Element of REAL
() . 0 is V25() real ext-real Element of REAL
x . 0 is V25() real ext-real Element of REAL
log (2,1) is V25() real ext-real Element of REAL
f is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
x . f is V25() real ext-real Element of REAL
Sum ((),f) is V25() real ext-real Element of REAL
c is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 * c is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
log (2,c) is V25() real ext-real Element of REAL
c * (log (2,c)) is V25() real ext-real Element of REAL
2 ^2 is V25() real ext-real Element of REAL
2 * 2 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer set
log (2,(2 ^2)) is V25() real ext-real Element of REAL
2 to_power 2 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 |^ 2 is set
log (2,(2 to_power 2)) is V25() real ext-real Element of REAL
log (2,2) is V25() real ext-real Element of REAL
2 * (log (2,2)) is V25() real ext-real Element of REAL
2 * 1 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
f is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() Element of K19(K20(NAT,REAL))
x is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() Element of K19(K20(NAT,REAL))
e is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
f . e is V25() real ext-real Element of REAL
log (2,2) is V25() real ext-real Element of REAL
log (2,e) is V25() real ext-real Element of REAL
e * (log (2,e)) is V25() real ext-real Element of REAL
e * 0 is 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
N0 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
N0 / 2 is V25() real ext-real non negative Element of REAL
N0 * (2 ") is V25() real ext-real non negative set
[/(N0 / 2)\] is V25() real ext-real integer set
log (2,(N0 / 2)) is V25() real ext-real Element of REAL
(N0 / 2) * (log (2,(N0 / 2))) is V25() real ext-real Element of REAL
log (2,N0) is V25() real ext-real Element of REAL
log (2,2) is V25() real ext-real Element of REAL
(log (2,N0)) - (log (2,2)) is V25() real ext-real Element of REAL
- (log (2,2)) is V25() real ext-real set
(log (2,N0)) + (- (log (2,2))) is V25() real ext-real set
(N0 / 2) * ((log (2,N0)) - (log (2,2))) is V25() real ext-real Element of REAL
(log (2,N0)) - 1 is V25() real ext-real Element of REAL
- 1 is V25() real ext-real non positive integer set
(log (2,N0)) + (- 1) is V25() real ext-real set
(N0 / 2) * ((log (2,N0)) - 1) is V25() real ext-real Element of REAL
N0 * (log (2,N0)) is V25() real ext-real Element of REAL
(N0 * (log (2,N0))) / 2 is V25() real ext-real Element of REAL
(N0 * (log (2,N0))) * (2 ") is V25() real ext-real set
((N0 * (log (2,N0))) / 2) - (N0 / 2) is V25() real ext-real Element of REAL
- (N0 / 2) is V25() real ext-real non positive set
((N0 * (log (2,N0))) / 2) + (- (N0 / 2)) is V25() real ext-real set
d is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
d is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() Element of K19(K20(NAT,REAL))
d . 0 is V25() real ext-real Element of REAL
N1 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
d . N1 is V25() real ext-real Element of REAL
d is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() Element of K19(K20(NAT,REAL))
d . 0 is V25() real ext-real Element of REAL
d is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() Element of K19(K20(NAT,REAL))
d . 0 is V25() real ext-real Element of REAL
N1 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
N1 - 1 is V25() real ext-real integer Element of REAL
N1 + (- 1) is V25() real ext-real integer set
2 " is non empty V25() real ext-real positive non negative Element of REAL
N0 * (2 ") is V25() real ext-real non negative Element of REAL
0 * (2 ") is functional empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V25() real ext-real non positive non negative integer V49() V50() V51() V52() V53() V54() V55() FinSequence-membered Element of REAL
(N1 - 1) + 1 is V25() real ext-real integer Element of REAL
(- 1) + 1 is V25() real ext-real integer Element of REAL
2 * N0 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
4 " is non empty V25() real ext-real positive non negative Element of REAL
(2 * N0) * (4 ") is V25() real ext-real non negative Element of REAL
(N0 * (log (2,N0))) * (4 ") is V25() real ext-real Element of REAL
(N0 * (log (2,N0))) / 4 is V25() real ext-real Element of REAL
4 " is non empty V25() real ext-real positive non negative set
(N0 * (log (2,N0))) * (4 ") is V25() real ext-real set
((N0 * (log (2,N0))) / 2) - ((N0 * (log (2,N0))) / 4) is V25() real ext-real Element of REAL
- ((N0 * (log (2,N0))) / 4) is V25() real ext-real set
((N0 * (log (2,N0))) / 2) + (- ((N0 * (log (2,N0))) / 4)) is V25() real ext-real set
(N0 / 2) + ((N0 * (log (2,N0))) / 4) is V25() real ext-real Element of REAL
2 * 2 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
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
N + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
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
d . n is V25() real ext-real Element of REAL
() . n is V25() real ext-real Element of REAL
log (2,n) is V25() real ext-real Element of REAL
Sum (d,N0,N) is V25() real ext-real Element of REAL
Sum ((),N0,N) is V25() real ext-real Element of REAL
(N0 / 2) + 1 is non empty V25() real ext-real positive non negative Element of REAL
(N0 / 2) + (N0 / 2) is V25() real ext-real non negative Element of REAL
(N0 / 2) + N is V25() real ext-real non negative Element of REAL
N0 - N is V25() real ext-real integer Element of REAL
- N is V25() real ext-real non positive integer set
N0 + (- N) is V25() real ext-real integer set
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
() . n is V25() real ext-real Element of REAL
0 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
log (2,1) is V25() real ext-real Element of REAL
log (2,n) is V25() real ext-real Element of REAL
Sum ((),N) is V25() real ext-real Element of REAL
Sum ((),N0) is V25() real ext-real Element of REAL
(Sum ((),N0)) + 0 is V25() real ext-real Element of REAL
(Sum ((),N0)) + (Sum ((),N)) is V25() real ext-real Element of REAL
(Sum ((),N0)) - (Sum ((),N)) is V25() real ext-real Element of REAL
- (Sum ((),N)) is V25() real ext-real set
(Sum ((),N0)) + (- (Sum ((),N))) is V25() real ext-real set
(N0 - N) * (log (2,(N0 / 2))) is V25() real ext-real Element of REAL
e is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() eventually-nonnegative eventually-positive eventually-nonzero Element of K19(K20(NAT,REAL))
e . N0 is V25() real ext-real Element of REAL
(e . N0) / 4 is V25() real ext-real Element of REAL
(e . N0) * (4 ") is V25() real ext-real set
1 / 4 is V25() real ext-real non negative Element of REAL
1 * (4 ") is V25() real ext-real non negative set
(1 / 4) * (e . N0) is V25() real ext-real Element of REAL
x . N0 is V25() real ext-real Element of REAL
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
n is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() Element of K19(K20(NAT,REAL))
n . 0 is V25() real ext-real Element of REAL
g9 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
n . g9 is V25() real ext-real Element of REAL
n is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() Element of K19(K20(NAT,REAL))
n . 0 is V25() real ext-real Element of REAL
n is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() Element of K19(K20(NAT,REAL))
n . 0 is V25() real ext-real Element of REAL
Sum (n,N0) is V25() real ext-real Element of REAL
g9 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
() . g9 is V25() real ext-real Element of REAL
n . g9 is V25() real ext-real Element of REAL
log (2,g9) is V25() real ext-real Element of REAL
N0 ! is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
log (2,(N0 !)) is V25() real ext-real Element of REAL
1 * (e . N0) is V25() real ext-real Element of REAL
Big_Theta e is functional non empty FUNCTION_DOMAIN of NAT , REAL
Big_Oh e is functional non empty FUNCTION_DOMAIN of NAT , REAL
{ b₁ where b₁ is V1() V4( NAT ) V5( REAL ) Function-like V14( NAT ) quasi_total V35() V36() V37() Element of Funcs (NAT,REAL) : ex b₂ being V25() real ext-real Element of REAL ex b₃ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT st
( not b₂ <= 0 & ( for b₄ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT holds
( not b₃ <= b₄ or ( b₁ . b₄ <= b₂ * (e . b₄) & 0 <= b₁ . b₄ ) ) ) ) } is set
Big_Omega e is functional non empty FUNCTION_DOMAIN of NAT , REAL
{ b₁ where b₁ is V1() V4( NAT ) V5( REAL ) Function-like V14( NAT ) quasi_total V35() V36() V37() Element of Funcs (NAT,REAL) : ex b₂ being V25() real ext-real Element of REAL ex b₃ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT st
( not b₂ <= 0 & ( for b₄ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT holds
( not b₃ <= b₄ or ( b₂ * (e . b₄) <= b₁ . b₄ & 0 <= b₁ . b₄ ) ) ) ) } is set
(Big_Oh e) /\ (Big_Omega e) is set
{ b₁ where b₁ is V1() V4( NAT ) V5( REAL ) Function-like V14( NAT ) quasi_total V35() V36() V37() Element of Funcs (NAT,REAL) : ex b₂, b₃ being V25() real ext-real Element of REAL ex b₄ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT st
( not b₂ <= 0 & not b₃ <= 0 & ( for b₅ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT holds
( not b₄ <= b₅ or ( b₃ * (e . b₅) <= b₁ . b₅ & b₁ . b₅ <= b₂ * (e . b₅) ) ) ) ) } is set
c is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() eventually-nonnegative eventually-nondecreasing Element of K19(K20(NAT,REAL))
Big_Theta c is functional non empty FUNCTION_DOMAIN of NAT , REAL
Big_Oh c is functional non empty FUNCTION_DOMAIN of NAT , REAL
{ b₁ where b₁ is V1() V4( NAT ) V5( REAL ) Function-like V14( NAT ) quasi_total V35() V36() V37() Element of Funcs (NAT,REAL) : ex b₂ being V25() real ext-real Element of REAL ex b₃ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT st
( not b₂ <= 0 & ( for b₄ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT holds
( not b₃ <= b₄ or ( b₁ . b₄ <= b₂ * (c . b₄) & 0 <= b₁ . b₄ ) ) ) ) } is set
Big_Omega c is functional non empty FUNCTION_DOMAIN of NAT , REAL
{ b₁ where b₁ is V1() V4( NAT ) V5( REAL ) Function-like V14( NAT ) quasi_total V35() V36() V37() Element of Funcs (NAT,REAL) : ex b₂ being V25() real ext-real Element of REAL ex b₃ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT st
( not b₂ <= 0 & ( for b₄ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT holds
( not b₃ <= b₄ or ( b₂ * (c . b₄) <= b₁ . b₄ & 0 <= b₁ . b₄ ) ) ) ) } is set
(Big_Oh c) /\ (Big_Omega c) is set
x is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() Element of K19(K20(NAT,REAL))
{ b₁ where b₁ is V1() V4( NAT ) V5( REAL ) Function-like V14( NAT ) quasi_total V35() V36() V37() Element of Funcs (NAT,REAL) : ex b₂, b₃ being V25() real ext-real Element of REAL ex b₄ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT st
( not b₂ <= 0 & not b₃ <= 0 & ( for b₅ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT holds
( not b₄ <= b₅ or ( b₃ * (c . b₅) <= b₁ . b₅ & b₁ . b₅ <= b₂ * (c . b₅) ) ) ) ) } is set
f is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
e is V1() V4( NAT ) V5( REAL ) Function-like V14( NAT ) quasi_total V35() V36() V37() Element of Funcs (NAT,REAL)
N0 is V25() real ext-real Element of REAL
t is V25() real ext-real Element of REAL
d is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
N0 is V25() real ext-real Element of REAL
t is V25() real ext-real Element of REAL
d is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
N0 / t is V25() real ext-real Element of REAL
t " is V25() real ext-real set
N0 * (t ") is V25() real ext-real set
[/(N0 / t)\] is V25() real ext-real integer set
[/(N0 / t)\] + 1 is V25() real ext-real integer Element of REAL
max (d,f) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
max (([/(N0 / t)\] + 1),(max (d,f))) is V25() real ext-real Element of REAL
b is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
b mod 2 is V25() real ext-real integer set
[/(N0 / t)\] + 0 is V25() real ext-real integer Element of REAL
N0 " is V25() real ext-real Element of REAL
b * (N0 ") is V25() real ext-real Element of REAL
(N0 ") * (N0 / t) is V25() real ext-real Element of REAL
b / N0 is V25() real ext-real Element of REAL
N0 " is V25() real ext-real set
b * (N0 ") is V25() real ext-real set
(N0 ") * N0 is V25() real ext-real Element of REAL
1 / t is V25() real ext-real Element of REAL
1 * (t ") is V25() real ext-real set
((N0 ") * N0) * (1 / t) is V25() real ext-real Element of REAL
1 * (1 / t) is V25() real ext-real Element of REAL
c . b is V25() real ext-real Element of REAL
b + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
c . (b + 1) is V25() real ext-real Element of REAL
e . b is V25() real ext-real Element of REAL
N0 * (c . b) is V25() real ext-real Element of REAL
t * (c . (b + 1)) is V25() real ext-real Element of REAL
t * (1 / t) is V25() real ext-real Element of REAL
b + 0 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(b + 1) mod 2 is V25() real ext-real integer set
1 mod 2 is V25() real ext-real integer set
1 + (1 mod 2) is V25() real ext-real integer Element of REAL
(1 + (1 mod 2)) mod 2 is V25() real ext-real integer set
1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(1 + 1) mod 2 is V25() real ext-real integer set
x . (b + 1) is V25() real ext-real Element of REAL
b is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
b mod 2 is V25() real ext-real integer set
b + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(b + 1) mod 2 is V25() real ext-real integer set
1 mod 2 is V25() real ext-real integer set
0 + (1 mod 2) is V25() real ext-real integer Element of REAL
(0 + (1 mod 2)) mod 2 is V25() real ext-real integer set
0 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(0 + 1) mod 2 is V25() real ext-real integer set
e . (b + 1) is V25() real ext-real Element of REAL
b + 0 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
c . (b + 1) is V25() real ext-real Element of REAL
(b + 1) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
c . ((b + 1) + 1) is V25() real ext-real Element of REAL
[/(N0 / t)\] + 0 is V25() real ext-real integer Element of REAL
N0 " is V25() real ext-real Element of REAL
(b + 1) * (N0 ") is V25() real ext-real Element of REAL
(N0 ") * (N0 / t) is V25() real ext-real Element of REAL
(b + 1) / N0 is V25() real ext-real Element of REAL
N0 " is V25() real ext-real set
(b + 1) * (N0 ") is V25() real ext-real set
(N0 ") * N0 is V25() real ext-real Element of REAL
1 / t is V25() real ext-real Element of REAL
1 * (t ") is V25() real ext-real set
((N0 ") * N0) * (1 / t) is V25() real ext-real Element of REAL
1 * (1 / t) is V25() real ext-real Element of REAL
N0 * (c . (b + 1)) is V25() real ext-real Element of REAL
b + 2 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
c . (b + 2) is V25() real ext-real Element of REAL
t * (c . (b + 2)) is V25() real ext-real Element of REAL
t * (1 / t) is V25() real ext-real Element of REAL
(b + 2) mod 2 is V25() real ext-real integer set
2 mod 2 is V25() real ext-real integer set
0 + (2 mod 2) is V25() real ext-real integer Element of REAL
(0 + (2 mod 2)) mod 2 is V25() real ext-real integer set
0 + 0 is 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
(0 + 0) mod 2 is V25() real ext-real integer set
x . (b + 2) is V25() real ext-real Element of REAL
b is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
b mod 2 is V25() real ext-real integer set
c is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
c / 2 is V25() real ext-real non negative Element of REAL
c * (2 ") is V25() real ext-real non negative set
[/(c / 2)\] is V25() real ext-real integer set
(c / 2) + 1 is non empty V25() real ext-real positive non negative Element of REAL
2 * ((c / 2) + 1) is V25() real ext-real non negative Element of REAL
2 * c is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 * (c / 2) is V25() real ext-real non negative Element of REAL
2 * 1 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(2 * (c / 2)) + (2 * 1) is V25() real ext-real non negative Element of REAL
(2 * c) - c is V25() real ext-real integer Element of REAL
- c is V25() real ext-real non positive integer set
(2 * c) + (- c) is V25() real ext-real integer set
{ (2 to_power b₁) where b₁ is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT : verum } is set
x is set
f is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 to_power f is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 |^ f is set
2 to_power 1 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 |^ 1 is set
() is non empty V49() V50() V51() V52() V53() V54() Element of K19(NAT)
c is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer set
c ^2 is V25() real ext-real set
c * c is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer set
c + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(c + 1) ^2 is V25() real ext-real set
(c + 1) * (c + 1) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer set
(c + 1) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 * c is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 * 0 is 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
(2 * c) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
0 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(c + 1) + ((2 * c) + 1) is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(c + 1) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(c ^2) + ((2 * c) + 1) is V25() real ext-real Element of REAL
(c + 1) ^2 is V25() real ext-real Element of REAL
2 ^2 is V25() real ext-real set
2 * 2 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer set
2 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
c is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
c ^2 is V25() real ext-real Element of REAL
c * c is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer set
c + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
c is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
c + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 to_power (c + 1) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 |^ (c + 1) is set
2 to_power c is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 |^ c is set
(2 to_power (c + 1)) - (2 to_power c) is V25() real ext-real integer Element of REAL
- (2 to_power c) is V25() real ext-real non positive integer set
(2 to_power (c + 1)) + (- (2 to_power c)) is V25() real ext-real integer set
2 to_power 1 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 |^ 1 is set
(2 to_power c) * 1 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(2 to_power c) * 2 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
((2 to_power c) * 2) - ((2 to_power c) * 1) is V25() real ext-real integer Element of REAL
- ((2 to_power c) * 1) is V25() real ext-real non positive integer set
((2 to_power c) * 2) + (- ((2 to_power c) * 1)) is V25() real ext-real integer set
(2 to_power c) * (2 to_power 1) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
((2 to_power c) * (2 to_power 1)) - (2 to_power c) is V25() real ext-real integer Element of REAL
((2 to_power c) * (2 to_power 1)) + (- (2 to_power c)) is V25() real ext-real integer set
2 - 1 is V25() real ext-real integer Element of REAL
- 1 is V25() real ext-real non positive integer set
2 + (- 1) is V25() real ext-real integer set
c is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 to_power c is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 |^ c is set
(2 to_power c) - 1 is V25() real ext-real integer Element of REAL
(2 to_power c) + (- 1) is V25() real ext-real integer set
c - 1 is V25() real ext-real integer Element of REAL
c + (- 1) is V25() real ext-real integer set
c + (- 1) is V25() real ext-real integer Element of REAL
(c + (- 1)) + 1 is V25() real ext-real integer Element of REAL
2 to_power ((c + (- 1)) + 1) is V25() real ext-real Element of REAL
2 to_power (c - 1) is V25() real ext-real Element of REAL
(2 to_power ((c + (- 1)) + 1)) - (2 to_power (c - 1)) is V25() real ext-real Element of REAL
- (2 to_power (c - 1)) is V25() real ext-real set
(2 to_power ((c + (- 1)) + 1)) + (- (2 to_power (c - 1))) is V25() real ext-real set
1 + (2 to_power (c - 1)) is V25() real ext-real Element of REAL
x is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 to_power x is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 |^ x is set
(2 to_power c) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(2 to_power c) + 0 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
x + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
Big_Theta ((1),()) is functional non empty FUNCTION_DOMAIN of NAT , REAL
{ b₁ where b₁ is V1() V4( NAT ) V5( REAL ) Function-like V14( NAT ) quasi_total V35() V36() V37() Element of Funcs (NAT,REAL) : ex b₂, b₃ being V25() real ext-real Element of REAL ex b₄ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT st
( not b₂ <= 0 & not b₃ <= 0 & ( for b₅ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT holds
( not b₄ <= b₅ or not b₅ in () or ( b₃ * ((1) . b₅) <= b₁ . b₅ & b₁ . b₅ <= b₂ * ((1) . b₅) ) ) ) ) } is set
Big_Theta (1) is functional non empty FUNCTION_DOMAIN of NAT , REAL
Big_Omega (1) is functional non empty FUNCTION_DOMAIN of NAT , REAL
{ b₁ where b₁ is V1() V4( NAT ) V5( REAL ) Function-like V14( NAT ) quasi_total V35() V36() V37() Element of Funcs (NAT,REAL) : ex b₂ being V25() real ext-real Element of REAL ex b₃ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT st
( not b₂ <= 0 & ( for b₄ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT holds
( not b₃ <= b₄ or ( b₂ * ((1) . b₄) <= b₁ . b₄ & 0 <= b₁ . b₄ ) ) ) ) } is set
(Big_Oh (1)) /\ (Big_Omega (1)) is set
(1) taken_every 2 is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() Element of K19(K20(NAT,REAL))
() /" (1) is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() Element of K19(K20(NAT,REAL))
(1) " is V1() V4( NAT ) Function-like V14( NAT ) V35() V36() V37() set
() (#) ((1) ") is V1() V4( NAT ) Function-like V14( NAT ) V35() V36() V37() set
t is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 * t is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 * 0 is 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
((1) taken_every 2) . t is V25() real ext-real Element of REAL
(1) . (2 * t) is V25() real ext-real Element of REAL
(2 * t) to_power 1 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(2 * t) |^ 1 is set
(1) . t is V25() real ext-real Element of REAL
t to_power 1 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
t |^ 1 is set
2 * ((1) . t) is V25() real ext-real Element of REAL
t is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() Element of K19(K20(NAT,REAL))
d is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(1) . d is V25() real ext-real Element of REAL
d to_power 1 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
d |^ 1 is set
1 * ((1) . d) is V25() real ext-real Element of REAL
t . d is V25() real ext-real Element of REAL
{ b₁ where b₁ is V1() V4( NAT ) V5( REAL ) Function-like V14( NAT ) quasi_total V35() V36() V37() Element of Funcs (NAT,REAL) : ex b₂, b₃ being V25() real ext-real Element of REAL ex b₄ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT st
( not b₂ <= 0 & not b₃ <= 0 & ( for b₅ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT holds
( not b₄ <= b₅ or ( b₃ * ((1) . b₅) <= b₁ . b₅ & b₁ . b₅ <= b₂ * ((1) . b₅) ) ) ) ) } is set
lim (() /" (1)) is V25() real ext-real Element of REAL
d is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
N1 is V1() V4( NAT ) V5( REAL ) Function-like V14( NAT ) quasi_total V35() V36() V37() Element of Funcs (NAT,REAL)
b is V25() real ext-real Element of REAL
N is V25() real ext-real Element of REAL
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
b is V25() real ext-real Element of REAL
N is V25() real ext-real Element of REAL
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
max (d,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
[/b\] is V25() real ext-real integer set
max ([/b\],2) is V25() real ext-real set
max ((max (d,n)),(max ([/b\],2))) is V25() real ext-real set
N0 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 to_power N0 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 |^ N0 is set
(2 to_power N0) - 1 is V25() real ext-real integer Element of REAL
- 1 is V25() real ext-real non positive integer set
(2 to_power N0) + (- 1) is V25() real ext-real integer set
0 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
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
2 to_power 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
2 |^ n is set
N1 . n is V25() real ext-real Element of REAL
N0 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(1) . n is V25() real ext-real Element of REAL
n to_power 1 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
n |^ 1 is set
b * n is V25() real ext-real Element of REAL
log (2,(2 to_power n)) is V25() real ext-real Element of REAL
log (2,(b * n)) is V25() real ext-real Element of REAL
log (2,2) is V25() real ext-real Element of REAL
n * (log (2,2)) is V25() real ext-real Element of REAL
n * 1 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
log (2,b) is V25() real ext-real Element of REAL
log (2,n) is V25() real ext-real Element of REAL
(log (2,b)) + (log (2,n)) is V25() real ext-real Element of REAL
(log (2,n)) + (log (2,n)) is V25() real ext-real Element of REAL
2 * (log (2,n)) is V25() real ext-real Element of REAL
n / 2 is V25() real ext-real non negative Element of REAL
n * (2 ") is V25() real ext-real non negative set
n " is V25() real ext-real non negative Element of REAL
n * (1 / 2) is V25() real ext-real non negative Element of REAL
(n ") * (n * (1 / 2)) is V25() real ext-real non negative Element of REAL
(log (2,n)) * (n ") is V25() real ext-real Element of REAL
(n ") * n is V25() real ext-real non negative Element of REAL
((n ") * n) * (1 / 2) is V25() real ext-real non negative Element of REAL
(log (2,n)) / n is V25() real ext-real Element of REAL
n " is V25() real ext-real non negative set
(log (2,n)) * (n ") is V25() real ext-real set
(() /" (1)) . n is V25() real ext-real Element of REAL
((() /" (1)) . n) - 0 is V25() real ext-real Element of REAL
- 0 is functional empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V25() real ext-real non positive non negative integer V49() V50() V51() V52() V53() V54() V55() FinSequence-membered set
((() /" (1)) . n) + (- 0) is V25() real ext-real set
abs (((() /" (1)) . n) - 0) is V25() real ext-real Element of REAL
() . n is V25() real ext-real Element of REAL
(() . n) / ((1) . n) is V25() real ext-real Element of REAL
((1) . n) " is V25() real ext-real set
(() . n) * (((1) . n) ") is V25() real ext-real set
(log (2,n)) / ((1) . n) is V25() real ext-real Element of REAL
(log (2,n)) * (((1) . n) ") is V25() real ext-real set
(log (2,n)) / (n to_power 1) is V25() real ext-real Element of REAL
(n to_power 1) " is V25() real ext-real non negative set
(log (2,n)) * ((n to_power 1) ") is V25() real ext-real set
d is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
d + 0 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
d + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(1) . d is V25() real ext-real Element of REAL
d to_power 1 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
d |^ 1 is set
(1) . (d + 1) is V25() real ext-real Element of REAL
(d + 1) to_power 1 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(d + 1) |^ 1 is set
4 - 1 is V25() real ext-real integer Element of REAL
4 + (- 1) is V25() real ext-real integer set
d is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
d + 2 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 to_power (d + 2) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 |^ (d + 2) is set
(2 to_power (d + 2)) - 1 is V25() real ext-real integer Element of REAL
(2 to_power (d + 2)) + (- 1) is V25() real ext-real integer set
2 to_power 2 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 |^ 2 is set
2 ^2 is V25() real ext-real Element of REAL
2 * 2 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer set
0 + 2 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(d + 2) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
b is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
d + 0 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
b + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
t . (b + 1) is V25() real ext-real Element of REAL
t . b is V25() real ext-real Element of REAL
2 to_power b is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 |^ b is set
x is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() Element of K19(K20(NAT,REAL))
f is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() Element of K19(K20(NAT,REAL))
e is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
f . e is V25() real ext-real Element of REAL
e to_power e is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
e |^ e is set
f taken_every 2 is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() Element of K19(K20(NAT,REAL))
t is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
d is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 to_power d is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 |^ d is set
x . t is V25() real ext-real Element of REAL
log (2,t) is V25() real ext-real Element of REAL
[\(log (2,t))/] is V25() real ext-real integer set
2 to_power [\(log (2,t))/] is V25() real ext-real set
t to_power (2 to_power [\(log (2,t))/]) is V25() real ext-real set
log (2,2) is V25() real ext-real Element of REAL
d * (log (2,2)) is V25() real ext-real Element of REAL
d * 1 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
t to_power t is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
t |^ t is set
N0 is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() eventually-nonnegative eventually-positive eventually-nonzero Element of K19(K20(NAT,REAL))
N0 . t is V25() real ext-real Element of REAL
1 * (N0 . t) is V25() real ext-real Element of REAL
Big_Theta N0 is functional non empty FUNCTION_DOMAIN of NAT , REAL
Big_Oh N0 is functional non empty FUNCTION_DOMAIN of NAT , REAL
{ b₁ where b₁ is V1() V4( NAT ) V5( REAL ) Function-like V14( NAT ) quasi_total V35() V36() V37() Element of Funcs (NAT,REAL) : ex b₂ being V25() real ext-real Element of REAL ex b₃ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT st
( not b₂ <= 0 & ( for b₄ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT holds
( not b₃ <= b₄ or ( b₁ . b₄ <= b₂ * (N0 . b₄) & 0 <= b₁ . b₄ ) ) ) ) } is set
Big_Omega N0 is functional non empty FUNCTION_DOMAIN of NAT , REAL
{ b₁ where b₁ is V1() V4( NAT ) V5( REAL ) Function-like V14( NAT ) quasi_total V35() V36() V37() Element of Funcs (NAT,REAL) : ex b₂ being V25() real ext-real Element of REAL ex b₃ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT st
( not b₂ <= 0 & ( for b₄ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT holds
( not b₃ <= b₄ or ( b₂ * (N0 . b₄) <= b₁ . b₄ & 0 <= b₁ . b₄ ) ) ) ) } is set
(Big_Oh N0) /\ (Big_Omega N0) is set
{ b₁ where b₁ is V1() V4( NAT ) V5( REAL ) Function-like V14( NAT ) quasi_total V35() V36() V37() Element of Funcs (NAT,REAL) : ex b₂, b₃ being V25() real ext-real Element of REAL ex b₄ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT st
( not b₂ <= 0 & not b₃ <= 0 & ( for b₅ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT holds
( not b₄ <= b₅ or ( b₃ * (N0 . b₅) <= b₁ . b₅ & b₁ . b₅ <= b₂ * (N0 . b₅) ) ) ) ) } is set
t is V1() V4( NAT ) V5( REAL ) Function-like V14( NAT ) quasi_total V35() V36() V37() Element of Funcs (NAT,REAL)
d is V25() real ext-real Element of REAL
N1 is V25() real ext-real Element of REAL
b is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
d is V25() real ext-real Element of REAL
N1 is V25() real ext-real Element of REAL
b is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
1 / N1 is V25() real ext-real Element of REAL
N1 " is V25() real ext-real set
1 * (N1 ") is V25() real ext-real set
[/(1 / N1)\] is V25() real ext-real integer set
max (b,2) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
max ([/(1 / N1)\],(max (b,2))) is V25() real ext-real set
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
2 to_power 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
2 |^ n is set
g9 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
n + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 to_power (n + 1) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 |^ (n + 1) is set
(2 to_power (n + 1)) - (2 to_power n) is V25() real ext-real integer Element of REAL
- (2 to_power n) is V25() real ext-real non positive integer set
(2 to_power (n + 1)) + (- (2 to_power n)) is V25() real ext-real integer set
g9 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
log (2,(2 to_power (n + 1))) is V25() real ext-real Element of REAL
log (2,(g9 + 1)) is V25() real ext-real Element of REAL
(n + 1) * (log (2,2)) is V25() real ext-real Element of REAL
(n + 1) * 1 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
[\(log (2,(g9 + 1)))/] is V25() real ext-real integer set
N0 . (g9 + 1) is V25() real ext-real Element of REAL
(g9 + 1) to_power (g9 + 1) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(g9 + 1) |^ (g9 + 1) is set
n + 0 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
g9 + 0 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
N1 * (N0 . (g9 + 1)) is V25() real ext-real Element of REAL
t . (g9 + 1) is V25() real ext-real Element of REAL
(1 / N1) + 0 is V25() real ext-real Element of REAL
log (2,g9) is V25() real ext-real Element of REAL
n * (log (2,2)) is V25() real ext-real Element of REAL
n * 1 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
[\n/] is V25() real ext-real integer set
2 to_power [\(log (2,(g9 + 1)))/] is V25() real ext-real set
(g9 + 1) to_power (2 to_power [\(log (2,(g9 + 1)))/]) is V25() real ext-real set
(N0 . (g9 + 1)) / (t . (g9 + 1)) is V25() real ext-real Element of REAL
(t . (g9 + 1)) " is V25() real ext-real set
(N0 . (g9 + 1)) * ((t . (g9 + 1)) ") is V25() real ext-real set
(g9 + 1) to_power g9 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(g9 + 1) |^ g9 is set
((g9 + 1) to_power (g9 + 1)) / ((g9 + 1) to_power g9) is V25() real ext-real non negative Element of REAL
((g9 + 1) to_power g9) " is V25() real ext-real non negative set
((g9 + 1) to_power (g9 + 1)) * (((g9 + 1) to_power g9) ") is V25() real ext-real non negative set
(g9 + 1) - g9 is V25() real ext-real integer Element of REAL
- g9 is V25() real ext-real non positive integer set
(g9 + 1) + (- g9) is V25() real ext-real integer set
(g9 + 1) to_power ((g9 + 1) - g9) is V25() real ext-real Element of REAL
1 / (1 / N1) is V25() real ext-real Element of REAL
(1 / N1) " is V25() real ext-real set
1 * ((1 / N1) ") is V25() real ext-real set
1 / ((N0 . (g9 + 1)) / (t . (g9 + 1))) is V25() real ext-real Element of REAL
((N0 . (g9 + 1)) / (t . (g9 + 1))) " is V25() real ext-real set
1 * (((N0 . (g9 + 1)) / (t . (g9 + 1))) ") is V25() real ext-real set
(t . (g9 + 1)) / (N0 . (g9 + 1)) is V25() real ext-real Element of REAL
(N0 . (g9 + 1)) " is V25() real ext-real set
(t . (g9 + 1)) * ((N0 . (g9 + 1)) ") is V25() real ext-real set
((t . (g9 + 1)) / (N0 . (g9 + 1))) * (N0 . (g9 + 1)) is V25() real ext-real Element of REAL
t is V1() V4( NAT ) V5( REAL ) Function-like V14( NAT ) quasi_total V35() V36() V37() Element of Funcs (NAT,REAL)
d is V25() real ext-real Element of REAL
N1 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
d is V25() real ext-real Element of REAL
N1 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
[/d\] is V25() real ext-real integer set
max (N1,2) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
max ([/d\],(max (N1,2))) is V25() real ext-real set
2 * (max ([/d\],(max (N1,2)))) is V25() real ext-real Element of REAL
1 * (max ([/d\],(max (N1,2)))) is V25() real ext-real Element of REAL
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
N + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
N + 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
N to_power (N + 1) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
N |^ (N + 1) is set
2 * 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
N to_power (2 * 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
N |^ (2 * N) is set
N to_power 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
N |^ N is set
N to_power 1 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
N |^ 1 is set
(N to_power N) * (N to_power 1) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(N to_power N) * 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
d * (N to_power N) is V25() real ext-real Element of REAL
(f taken_every 2) . N is V25() real ext-real Element of REAL
N0 . (2 * N) is V25() real ext-real Element of REAL
(2 * N) to_power (2 * 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
(2 * N) |^ (2 * N) is set
N0 . N is V25() real ext-real Element of REAL
d * (N0 . N) is V25() real ext-real Element of REAL
t is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
t + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
x . (t + 1) is V25() real ext-real Element of REAL
log (2,(t + 1)) is V25() real ext-real Element of REAL
[\(log (2,(t + 1)))/] is V25() real ext-real integer set
2 to_power [\(log (2,(t + 1)))/] is V25() real ext-real set
(t + 1) to_power (2 to_power [\(log (2,(t + 1)))/]) is V25() real ext-real set
x . t is V25() real ext-real Element of REAL
log (2,t) is V25() real ext-real Element of REAL
[\(log (2,t))/] is V25() real ext-real integer set
2 to_power [\(log (2,t))/] is V25() real ext-real set
t to_power (2 to_power [\(log (2,t))/]) is V25() real ext-real set
t + 0 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
0 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(t + 1) to_power (2 to_power [\(log (2,t))/]) is V25() real ext-real set
[\1/] is V25() real ext-real integer set
2 to_power 0 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 |^ 0 is set
t is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
t + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
t + 0 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(t + 1) to_power t is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(t + 1) |^ t is set
t to_power t is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
t |^ t is set
1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(t + 1) to_power (t + 1) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(t + 1) |^ (t + 1) is set
N0 . (t + 1) is V25() real ext-real Element of REAL
N0 . t is V25() real ext-real Element of REAL
Big_Theta (N0,()) is functional non empty FUNCTION_DOMAIN of NAT , REAL
{ b₁ where b₁ is V1() V4( NAT ) V5( REAL ) Function-like V14( NAT ) quasi_total V35() V36() V37() Element of Funcs (NAT,REAL) : ex b₂, b₃ being V25() real ext-real Element of REAL ex b₄ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT st
( not b₂ <= 0 & not b₃ <= 0 & ( for b₅ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT holds
( not b₄ <= b₅ or not b₅ in () or ( b₃ * (N0 . b₅) <= b₁ . b₅ & b₁ . b₅ <= b₂ * (N0 . b₅) ) ) ) ) } is set
c is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() Element of K19(K20(NAT,REAL))
x is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
c . x is V25() real ext-real Element of REAL
x to_power 2 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
x |^ 2 is set
c taken_every 2 is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() Element of K19(K20(NAT,REAL))
f is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() eventually-nonnegative eventually-positive eventually-nonzero Element of K19(K20(NAT,REAL))
Big_Theta (f,()) is functional non empty FUNCTION_DOMAIN of NAT , REAL
{ b₁ where b₁ is V1() V4( NAT ) V5( REAL ) Function-like V14( NAT ) quasi_total V35() V36() V37() Element of Funcs (NAT,REAL) : ex b₂, b₃ being V25() real ext-real Element of REAL ex b₄ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT st
( not b₂ <= 0 & not b₃ <= 0 & ( for b₅ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT holds
( not b₄ <= b₅ or not b₅ in () or ( b₃ * (f . b₅) <= b₁ . b₅ & b₁ . b₅ <= b₂ * (f . b₅) ) ) ) ) } is set
Big_Theta f is functional non empty FUNCTION_DOMAIN of NAT , REAL
Big_Oh f is functional non empty FUNCTION_DOMAIN of NAT , REAL
{ b₁ where b₁ is V1() V4( NAT ) V5( REAL ) Function-like V14( NAT ) quasi_total V35() V36() V37() Element of Funcs (NAT,REAL) : ex b₂ being V25() real ext-real Element of REAL ex b₃ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT st
( not b₂ <= 0 & ( for b₄ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT holds
( not b₃ <= b₄ or ( b₁ . b₄ <= b₂ * (f . b₄) & 0 <= b₁ . b₄ ) ) ) ) } is set
Big_Omega f is functional non empty FUNCTION_DOMAIN of NAT , REAL
{ b₁ where b₁ is V1() V4( NAT ) V5( REAL ) Function-like V14( NAT ) quasi_total V35() V36() V37() Element of Funcs (NAT,REAL) : ex b₂ being V25() real ext-real Element of REAL ex b₃ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT st
( not b₂ <= 0 & ( for b₄ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT holds
( not b₃ <= b₄ or ( b₂ * (f . b₄) <= b₁ . b₄ & 0 <= b₁ . b₄ ) ) ) ) } is set
(Big_Oh f) /\ (Big_Omega f) is set
f taken_every 2 is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() Element of K19(K20(NAT,REAL))
t is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(c taken_every 2) . t is V25() real ext-real Element of REAL
2 * t is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
f . (2 * t) is V25() real ext-real Element of REAL
f . t is V25() real ext-real Element of REAL
4 * (f . t) is V25() real ext-real Element of REAL
d is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 to_power d is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 |^ d is set
2 to_power 1 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 |^ 1 is set
(2 to_power 1) * (2 to_power d) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
d + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 to_power (d + 1) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 |^ (d + 1) is set
d is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 to_power d is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 |^ d is set
0 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
t * 2 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 " is non empty V25() real ext-real positive non negative Element of REAL
(t * 2) * (2 ") is V25() real ext-real non negative Element of REAL
1 * (2 ") is V25() real ext-real non negative Element of REAL
0 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
d - 1 is V25() real ext-real integer Element of REAL
- 1 is V25() real ext-real non positive integer set
d + (- 1) is V25() real ext-real integer set
d + (- 1) is V25() real ext-real integer Element of REAL
(d + (- 1)) + 1 is V25() real ext-real integer Element of REAL
2 to_power ((d + (- 1)) + 1) is V25() real ext-real Element of REAL
2 to_power (d - 1) is V25() real ext-real Element of REAL
2 to_power 1 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 |^ 1 is set
(2 to_power (d - 1)) * (2 to_power 1) is V25() real ext-real Element of REAL
(2 to_power (d - 1)) * 2 is V25() real ext-real Element of REAL
(2 * t) to_power 2 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(2 * t) |^ 2 is set
(2 * t) ^2 is V25() real ext-real Element of REAL
(2 * t) * (2 * t) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer set
t ^2 is V25() real ext-real Element of REAL
t * t is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer set
4 * (t ^2) is V25() real ext-real Element of REAL
t to_power 2 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
t |^ 2 is set
(2 * t) to_power 2 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(2 * t) |^ 2 is set
{ b₁ where b₁ is V1() V4( NAT ) V5( REAL ) Function-like V14( NAT ) quasi_total V35() V36() V37() Element of Funcs (NAT,REAL) : ex b₂, b₃ being V25() real ext-real Element of REAL ex b₄ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT st
( not b₂ <= 0 & not b₃ <= 0 & ( for b₅ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT holds
( not b₄ <= b₅ or ( b₃ * (f . b₅) <= b₁ . b₅ & b₁ . b₅ <= b₂ * (f . b₅) ) ) ) ) } is set
t is V1() V4( NAT ) V5( REAL ) Function-like V14( NAT ) quasi_total V35() V36() V37() Element of Funcs (NAT,REAL)
d is V25() real ext-real Element of REAL
N1 is V25() real ext-real Element of REAL
b is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
d is V25() real ext-real Element of REAL
N1 is V25() real ext-real Element of REAL
b is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
max (b,2) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
1 / N1 is V25() real ext-real Element of REAL
N1 " is V25() real ext-real set
1 * (N1 ") is V25() real ext-real set
[/(1 / N1)\] is V25() real ext-real integer set
max ((max (b,2)),[/(1 / N1)\]) is V25() real ext-real set
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
2 to_power 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
2 |^ n is set
(2 to_power n) - 1 is V25() real ext-real integer Element of REAL
- 1 is V25() real ext-real non positive integer set
(2 to_power n) + (- 1) is V25() real ext-real integer set
0 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
N0 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
f . N0 is V25() real ext-real Element of REAL
N0 to_power 2 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
N0 |^ 2 is set
N0 ^2 is V25() real ext-real Element of REAL
N0 * N0 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer set
n + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
N0 * N0 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
N0 * (1 / N1) is V25() real ext-real Element of REAL
N1 * (N0 ^2) is V25() real ext-real Element of REAL
(N0 * (1 / N1)) * N1 is V25() real ext-real Element of REAL
(1 / N1) * N1 is V25() real ext-real Element of REAL
N0 * ((1 / N1) * N1) is V25() real ext-real Element of REAL
N0 * 1 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
t . N0 is V25() real ext-real Element of REAL
N0 to_power 1 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
N0 |^ 1 is set
4 - 1 is V25() real ext-real integer Element of REAL
4 + (- 1) is V25() real ext-real integer set
t is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
max (t,1) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 * (max (t,1)) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 to_power (2 * (max (t,1))) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 |^ (2 * (max (t,1))) is set
(2 to_power (2 * (max (t,1)))) - 1 is V25() real ext-real integer Element of REAL
(2 to_power (2 * (max (t,1)))) + (- 1) is V25() real ext-real integer set
2 to_power 0 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 |^ 0 is set
1 - 1 is V25() real ext-real integer Element of REAL
1 + (- 1) is V25() real ext-real integer set
2 * 1 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(2 * (max (t,1))) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
b is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 to_power 2 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 |^ 2 is set
2 ^2 is V25() real ext-real Element of REAL
2 * 2 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer set
b ^2 is V25() real ext-real Element of REAL
b * b is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer set
b + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
f . b is V25() real ext-real Element of REAL
b to_power 2 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
b |^ 2 is set
1 * (max (t,1)) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
f . (b + 1) is V25() real ext-real Element of REAL
t is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
t + 0 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
t + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(1) . t is V25() real ext-real Element of REAL
t to_power 1 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
t |^ 1 is set
(1) . (t + 1) is V25() real ext-real Element of REAL
(t + 1) to_power 1 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(t + 1) |^ 1 is set
t is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(1) . t is V25() real ext-real Element of REAL
t to_power 1 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
t |^ 1 is set
f . t is V25() real ext-real Element of REAL
1 * (f . t) is V25() real ext-real Element of REAL
c is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer set
c ! is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
c + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(c + 1) ! is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
0 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(c + 1) * (c !) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(c + 1) * 1 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 ! is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
c is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
c ! is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
c is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
c ! is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
c + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(c + 1) ! is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
x is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
x ! is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
0 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
1 * (c !) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(c + 1) * (c !) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
0 ! is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
c is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
c ! is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
x is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
c is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
x ! is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
c ! is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
c is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer set
c + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
x is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
x ! is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
x + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(x + 1) ! is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
x is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
x ! is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
x + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(x + 1) ! is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
c + 0 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
f is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
f ! is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
f + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(f + 1) ! is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
f is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
f ! is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
f + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(f + 1) ! is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
0 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(f + 1) * (f !) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
1 * (f !) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
e is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
e ! is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
e + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(e + 1) ! is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
0 + 2 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
x + 2 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
1 ! is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(1 + 1) ! is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
c is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
c ! is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
c + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(c + 1) ! is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
0 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
c is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
c is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
0 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
x is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
x ! is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
x + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(x + 1) ! is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
f is V25() real ext-real Element of REAL
e is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
N0 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
x is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
f is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
e is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
e ! is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
e + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(e + 1) ! is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
e is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
e ! is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
e + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(e + 1) ! is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
N0 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
N0 ! is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
N0 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(N0 + 1) ! is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
N0 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
N0 ! is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
N0 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(N0 + 1) ! is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
e is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
e ! is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
e + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(e + 1) ! is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
N0 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
N0 ! is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
N0 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(N0 + 1) ! is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
c is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
c ! is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
c - 1 is V25() real ext-real integer Element of REAL
- 1 is V25() real ext-real non positive integer set
c + (- 1) is V25() real ext-real integer set
3 - 1 is V25() real ext-real integer Element of REAL
3 + (- 1) is V25() real ext-real integer set
f is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
f ! is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
c * (f !) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
c * 1 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
f + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
x is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() Element of K19(K20(NAT,REAL))
f is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
x . f is V25() real ext-real Element of REAL
(f) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
e is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
e ! is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
e + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(e + 1) ! is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
f is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() eventually-nonnegative eventually-positive eventually-nonzero Element of K19(K20(NAT,REAL))
e is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
f . e is V25() real ext-real Element of REAL
e + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
f . (e + 1) is V25() real ext-real Element of REAL
0 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
f . (0 + 1) is V25() real ext-real Element of REAL
(1) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
f . 0 is V25() real ext-real Element of REAL
(0) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(e) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
N0 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
N0 ! is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
N0 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(N0 + 1) ! is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
((e + 1)) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
N0 + 2 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
0 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(N0 + 2) * ((N0 + 1) !) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
1 * ((N0 + 1) !) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(N0 + 1) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
((N0 + 1) + 1) ! is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
((e + 1)) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
e is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
f . e is V25() real ext-real Element of REAL
e + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
f . (e + 1) is V25() real ext-real Element of REAL
2 - 1 is V25() real ext-real integer Element of REAL
- 1 is V25() real ext-real non positive integer set
2 + (- 1) is V25() real ext-real integer set
f taken_every 2 is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() Element of K19(K20(NAT,REAL))
Big_Oh f is functional non empty FUNCTION_DOMAIN of NAT , REAL
{ b₁ where b₁ is V1() V4( NAT ) V5( REAL ) Function-like V14( NAT ) quasi_total V35() V36() V37() Element of Funcs (NAT,REAL) : ex b₂ being V25() real ext-real Element of REAL ex b₃ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT st
( not b₂ <= 0 & ( for b₄ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT holds
( not b₃ <= b₄ or ( b₁ . b₄ <= b₂ * (f . b₄) & 0 <= b₁ . b₄ ) ) ) ) } is set
N0 is V1() V4( NAT ) V5( REAL ) Function-like V14( NAT ) quasi_total V35() V36() V37() Element of Funcs (NAT,REAL)
t is V25() real ext-real Element of REAL
d is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
t is V25() real ext-real Element of REAL
d is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
max (d,3) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
[/t\] is V25() real ext-real integer set
[/t\] + 1 is V25() real ext-real integer Element of REAL
max ((max (d,3)),([/t\] + 1)) is V25() real ext-real Element of REAL
b is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
b ! is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(b !) - 1 is V25() real ext-real integer Element of REAL
(b !) + (- 1) is V25() real ext-real integer set
b - 1 is V25() real ext-real integer Element of REAL
b + (- 1) is V25() real ext-real integer set
1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
g9 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
N0 . g9 is V25() real ext-real Element of REAL
2 * g9 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
f . (2 * g9) is V25() real ext-real Element of REAL
b + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
f . g9 is V25() real ext-real Element of REAL
t * (f . g9) is V25() real ext-real Element of REAL
(b + 1) * (b !) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
b * (b !) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(b + 1) ! is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(b !) + 2 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(b !) + (b !) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 * (b !) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 * 1 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(2 * (b !)) - (2 * 1) is V25() real ext-real integer Element of REAL
- (2 * 1) is V25() real ext-real non positive integer set
(2 * (b !)) + (- (2 * 1)) is V25() real ext-real integer set
(b !) - 0 is V25() real ext-real non negative integer Element of REAL
- 0 is functional empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V25() real ext-real non positive non negative integer V49() V50() V51() V52() V53() V54() V55() FinSequence-membered set
(b !) + (- 0) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer set
N0 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
N0 ! is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
1 * 1 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(b - 1) * (N0 !) is V25() real ext-real integer Element of REAL
b * 1 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
((b - 1) * (N0 !)) * b is V25() real ext-real integer Element of REAL
N0 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(N0 !) * (N0 + 1) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(b - 1) * ((N0 !) * (N0 + 1)) is V25() real ext-real integer Element of REAL
(b - 1) * (b !) is V25() real ext-real integer Element of REAL
(b !) + b is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(b !) * 1 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
((b !) * 1) + ((b - 1) * (b !)) is V25() real ext-real integer Element of REAL
b + 0 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
b * (N0 !) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(b * (b !)) - b is V25() real ext-real integer Element of REAL
- b is V25() real ext-real non positive integer set
(b * (b !)) + (- b) is V25() real ext-real integer set
b * ((b !) - 1) is V25() real ext-real integer Element of REAL
(b * ((b !) - 1)) / (b * 1) is V25() real ext-real Element of REAL
(b * 1) " is V25() real ext-real non negative set
(b * ((b !) - 1)) * ((b * 1) ") is V25() real ext-real set
((b !) - 1) / 1 is V25() real ext-real Element of REAL
1 " is non empty V25() real ext-real positive non negative set
((b !) - 1) * (1 ") is V25() real ext-real set
[/t\] + 0 is V25() real ext-real integer Element of REAL
((2 * g9)) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(N0 + 1) ! is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(g9) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
e is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
f . e is V25() real ext-real Element of REAL
(1) . e is V25() real ext-real Element of REAL
f . 0 is V25() real ext-real Element of REAL
(0) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
e to_power 1 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
e |^ 1 is set
(e) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
N0 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
N0 ! is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
N0 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(N0 + 1) ! is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
e is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
x . e is V25() real ext-real Element of REAL
(1) . e is V25() real ext-real Element of REAL
(1) - (1) is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() Element of K19(K20(NAT,REAL))
- (1) is V1() V4( NAT ) Function-like V14( NAT ) V35() V36() V37() set
- 1 is V25() real ext-real non positive integer set
(- 1) (#) (1) is V1() V4( NAT ) Function-like V14( NAT ) V35() V36() V37() set
(1) + (- (1)) is V1() V4( NAT ) Function-like V14( NAT ) V35() V36() V37() set
f is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
((1) - (1)) . f is V25() real ext-real Element of REAL
(1) . f is V25() real ext-real Element of REAL
1 + 0 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(1) . f is V25() real ext-real Element of REAL
f to_power 1 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
f |^ 1 is set
- (1) is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() Element of K19(K20(NAT,REAL))
(- (1)) . f is V25() real ext-real Element of REAL
((1) . f) + ((- (1)) . f) is V25() real ext-real Element of REAL
f + (- 1) is V25() real ext-real integer Element of REAL
f - 1 is V25() real ext-real integer Element of REAL
f + (- 1) is V25() real ext-real integer set
{ b₁ where b₁ is V1() V4( NAT ) V5( REAL ) Function-like V14( NAT ) quasi_total V35() V36() V37() Element of Funcs (NAT,REAL) : ex b₂, b₃ being V25() real ext-real Element of REAL ex b₄ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT st
( not b₂ <= 0 & not b₃ <= 0 & ( for b₅ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT holds
( not b₄ <= b₅ or ( b₃ * ((1) . b₅) <= b₁ . b₅ & b₁ . b₅ <= b₂ * ((1) . b₅) ) ) ) ) } is set
N0 is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() eventually-nonnegative Element of K19(K20(NAT,REAL))
Big_Theta N0 is functional non empty FUNCTION_DOMAIN of NAT , REAL
Big_Oh N0 is functional non empty FUNCTION_DOMAIN of NAT , REAL
{ b₁ where b₁ is V1() V4( NAT ) V5( REAL ) Function-like V14( NAT ) quasi_total V35() V36() V37() Element of Funcs (NAT,REAL) : ex b₂ being V25() real ext-real Element of REAL ex b₃ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT st
( not b₂ <= 0 & ( for b₄ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT holds
( not b₃ <= b₄ or ( b₁ . b₄ <= b₂ * (N0 . b₄) & 0 <= b₁ . b₄ ) ) ) ) } is set
Big_Omega N0 is functional non empty FUNCTION_DOMAIN of NAT , REAL
{ b₁ where b₁ is V1() V4( NAT ) V5( REAL ) Function-like V14( NAT ) quasi_total V35() V36() V37() Element of Funcs (NAT,REAL) : ex b₂ being V25() real ext-real Element of REAL ex b₃ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT st
( not b₂ <= 0 & ( for b₄ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT holds
( not b₃ <= b₄ or ( b₂ * (N0 . b₄) <= b₁ . b₄ & 0 <= b₁ . b₄ ) ) ) ) } is set
(Big_Oh N0) /\ (Big_Omega N0) is set
(Big_Theta N0) + (Big_Theta (1)) is functional non empty FUNCTION_DOMAIN of NAT , REAL
{ b₁ where b₁ is V1() V4( NAT ) V5( REAL ) Function-like V14( NAT ) quasi_total V35() V36() V37() Element of Funcs (NAT,REAL) : ex b₂, b₃ being V1() V4( NAT ) V5( REAL ) Function-like V14( NAT ) quasi_total V35() V36() V37() Element of Funcs (NAT,REAL) st
( b₂ in Big_Theta N0 & b₃ in Big_Theta (1) & ( for b₄ being Element of NAT holds b₁ . b₄ = (b₂ . b₄) + (b₃ . b₄) ) ) } is set
{ b₁ where b₁ is V1() V4( NAT ) V5( REAL ) Function-like V14( NAT ) quasi_total V35() V36() V37() Element of Funcs (NAT,REAL) : ex b₂, b₃ being V25() real ext-real Element of REAL ex b₄ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT st
( not b₂ <= 0 & not b₃ <= 0 & ( for b₅ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT holds
( not b₄ <= b₅ or ( b₃ * (((1) - (1)) . b₅) <= b₁ . b₅ & b₁ . b₅ <= b₂ * (((1) - (1)) . b₅) ) ) ) ) } is set
t is set
d is V1() V4( NAT ) V5( REAL ) Function-like V14( NAT ) quasi_total V35() V36() V37() Element of Funcs (NAT,REAL)
N1 is V1() V4( NAT ) V5( REAL ) Function-like V14( NAT ) quasi_total V35() V36() V37() Element of Funcs (NAT,REAL)
b is V1() V4( NAT ) V5( REAL ) Function-like V14( NAT ) quasi_total V35() V36() V37() Element of Funcs (NAT,REAL)
N1 is V1() V4( NAT ) V5( REAL ) Function-like V14( NAT ) quasi_total V35() V36() V37() Element of Funcs (NAT,REAL)
b is V1() V4( NAT ) V5( REAL ) Function-like V14( NAT ) quasi_total V35() V36() V37() Element of Funcs (NAT,REAL)
N is V1() V4( NAT ) V5( REAL ) Function-like V14( NAT ) quasi_total V35() V36() V37() Element of Funcs (NAT,REAL)
n is V25() real ext-real Element of REAL
g9 is V25() real ext-real Element of REAL
N0 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
N is V25() real ext-real Element of REAL
n is V25() real ext-real Element of REAL
g9 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
N0 is V1() V4( NAT ) V5( REAL ) Function-like V14( NAT ) quasi_total V35() V36() V37() Element of Funcs (NAT,REAL)
N0 is V25() real ext-real Element of REAL
n is V25() real ext-real Element of REAL
d is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
N0 is V25() real ext-real Element of REAL
N0 is V25() real ext-real Element of REAL
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
N + N0 is V25() real ext-real Element of REAL
max (g9,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
max (1,(max (g9,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
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
(1) . n is V25() real ext-real Element of REAL
(1) . n is V25() real ext-real Element of REAL
n to_power 1 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
n |^ 1 is set
- n is V25() real ext-real non positive integer Element of REAL
(- n) * n is V25() real ext-real Element of REAL
(- 1) * n is V25() real ext-real Element of REAL
- n is V25() real ext-real Element of REAL
n * (- n) is V25() real ext-real Element of REAL
n + N0 is V25() real ext-real Element of REAL
(n + N0) * n is V25() real ext-real Element of REAL
(n * (- n)) + ((n + N0) * n) is V25() real ext-real Element of REAL
(- n) + ((n + N0) * n) is V25() real ext-real Element of REAL
((1) - (1)) . n is V25() real ext-real Element of REAL
- (1) is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() Element of K19(K20(NAT,REAL))
(- (1)) . n is V25() real ext-real Element of REAL
((1) . n) + ((- (1)) . n) is V25() real ext-real Element of REAL
- ((1) . n) is V25() real ext-real Element of REAL
((1) . n) + (- ((1) . n)) is V25() real ext-real Element of REAL
(n to_power 1) + (- ((1) . n)) is V25() real ext-real Element of REAL
n + (- 1) is V25() real ext-real integer Element of REAL
N0 * ((1) . n) is V25() real ext-real Element of REAL
b . n is V25() real ext-real Element of REAL
n * (((1) - (1)) . n) is V25() real ext-real Element of REAL
(n * (((1) - (1)) . n)) + (N0 * ((1) . n)) is V25() real ext-real Element of REAL
(n * (((1) - (1)) . n)) + (b . n) is V25() real ext-real Element of REAL
N0 * ((1) . n) is V25() real ext-real Element of REAL
N * (((1) - (1)) . n) is V25() real ext-real Element of REAL
(N * (((1) - (1)) . n)) + (b . n) is V25() real ext-real Element of REAL
(N * (((1) - (1)) . n)) + (N0 * ((1) . n)) is V25() real ext-real Element of REAL
N1 . n is V25() real ext-real Element of REAL
(N1 . n) + (b . n) is V25() real ext-real Element of REAL
N0 * n is V25() real ext-real Element of REAL
d . n is V25() real ext-real Element of REAL
- N is V25() real ext-real Element of REAL
(N + N0) * n is V25() real ext-real Element of REAL
(- N) + ((N + N0) * n) is V25() real ext-real Element of REAL
0 + ((N + N0) * n) is V25() real ext-real Element of REAL
(N + N0) * ((1) . n) is V25() real ext-real Element of REAL
d is V1() V4( NAT ) V5( REAL ) Function-like V14( NAT ) quasi_total V35() V36() V37() Element of Funcs (NAT,REAL)
N1 is V25() real ext-real Element of REAL
b is V25() real ext-real Element of REAL
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
N1 is V25() real ext-real Element of REAL
b is V25() real ext-real Element of REAL
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
2 " is non empty V25() real ext-real positive non negative Element of REAL
(2 ") (#) d is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() Element of K19(K20(NAT,REAL))
N0 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
d . N0 is V25() real ext-real Element of REAL
((2 ") (#) d) . N0 is V25() real ext-real Element of REAL
(((2 ") (#) d) . N0) + (((2 ") (#) d) . N0) is V25() real ext-real Element of REAL
(2 ") * (d . N0) is V25() real ext-real Element of REAL
(2 ") * b is V25() real ext-real Element of REAL
(2 ") * 0 is functional empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V25() real ext-real non positive non negative integer V49() V50() V51() V52() V53() V54() V55() FinSequence-membered Element of REAL
max (N,2) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
N0 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
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
d . n is V25() real ext-real Element of REAL
(2 ") * (d . n) is V25() real ext-real Element of REAL
(1) . n is V25() real ext-real Element of REAL
N1 * ((1) . n) is V25() real ext-real Element of REAL
(2 ") * (N1 * ((1) . n)) is V25() real ext-real Element of REAL
b * ((1) . n) is V25() real ext-real Element of REAL
(2 ") * (b * ((1) . n)) is V25() real ext-real Element of REAL
((2 ") * b) * ((1) . n) is V25() real ext-real Element of REAL
((2 ") (#) d) . n is V25() real ext-real Element of REAL
(2 ") * N1 is V25() real ext-real Element of REAL
((2 ") * N1) * ((1) . n) is V25() real ext-real Element of REAL
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
(1) . n is V25() real ext-real Element of REAL
(1) . n is V25() real ext-real Element of REAL
n to_power 1 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
n |^ 1 is set
(n to_power 1) - 0 is V25() real ext-real non negative integer Element of REAL
- 0 is functional empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V25() real ext-real non positive non negative integer V49() V50() V51() V52() V53() V54() V55() FinSequence-membered set
(n to_power 1) + (- 0) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer set
n - 0 is V25() real ext-real non negative integer Element of REAL
n + (- 0) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer set
n + 2 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
n + 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
2 * 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
2 * 1 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(2 * n) - (2 * 1) is V25() real ext-real integer Element of REAL
- (2 * 1) is V25() real ext-real non positive integer set
(2 * n) + (- (2 * 1)) is V25() real ext-real integer set
(2 ") * n is V25() real ext-real non negative Element of REAL
n - 1 is V25() real ext-real integer Element of REAL
n + (- 1) is V25() real ext-real integer set
2 * (n - 1) is V25() real ext-real integer Element of REAL
(2 ") * (2 * (n - 1)) is V25() real ext-real Element of REAL
N1 * ((2 ") * n) is V25() real ext-real Element of REAL
N1 * (n - 1) is V25() real ext-real Element of REAL
b * ((1) . n) is V25() real ext-real Element of REAL
(2 ") * (b * ((1) . n)) is V25() real ext-real Element of REAL
d . n is V25() real ext-real Element of REAL
(2 ") * (d . n) is V25() real ext-real Element of REAL
((1) - (1)) . n is V25() real ext-real Element of REAL
(- (1)) . n is V25() real ext-real Element of REAL
((1) . n) + ((- (1)) . n) is V25() real ext-real Element of REAL
((1) . n) + (- 1) is V25() real ext-real Element of REAL
((1) . n) - 1 is V25() real ext-real Element of REAL
((1) . n) + (- 1) is V25() real ext-real set
(n to_power 1) - 1 is V25() real ext-real integer Element of REAL
(n to_power 1) + (- 1) is V25() real ext-real integer set
((2 ") * b) * (((1) - (1)) . n) is V25() real ext-real Element of REAL
((2 ") * b) * ((1) . n) is V25() real ext-real Element of REAL
((2 ") (#) d) . n is V25() real ext-real Element of REAL
N1 * ((1) . n) is V25() real ext-real Element of REAL
(2 ") * (N1 * ((1) . n)) is V25() real ext-real Element of REAL
N1 * (((1) - (1)) . n) is V25() real ext-real Element of REAL
((- 1)) is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() eventually-nonnegative eventually-positive eventually-nonzero Element of K19(K20(NAT,REAL))
x is functional non empty FUNCTION_DOMAIN of NAT , REAL
x to_power (Big_Oh (1)) is functional non empty FUNCTION_DOMAIN of NAT , REAL
{ b₁ where b₁ is V1() V4( NAT ) V5( REAL ) Function-like V14( NAT ) quasi_total V35() V36() V37() Element of Funcs (NAT,REAL) : ex b₂, b₃ being V1() V4( NAT ) V5( REAL ) Function-like V14( NAT ) quasi_total V35() V36() V37() Element of Funcs (NAT,REAL) ex b₄ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT st
( b₂ in x & b₃ in Big_Oh (1) & ( for b₅ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT holds
( not b₄ <= b₅ or b₁ . b₅ = (b₂ . b₅) to_power (b₃ . b₅) ) ) ) } is set
f is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
((- 1)) . f is V25() real ext-real Element of REAL
(1) . f is V25() real ext-real Element of REAL
f is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
0 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
f to_power (- 1) is V25() real ext-real Element of REAL
f to_power 1 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
f |^ 1 is set
((- 1)) . f is V25() real ext-real Element of REAL
(1) . f is V25() real ext-real Element of REAL
f is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
f is functional non empty FUNCTION_DOMAIN of NAT , REAL
f to_power (Big_Oh (1)) is functional non empty FUNCTION_DOMAIN of NAT , REAL
{ b₁ where b₁ is V1() V4( NAT ) V5( REAL ) Function-like V14( NAT ) quasi_total V35() V36() V37() Element of Funcs (NAT,REAL) : ex b₂, b₃ being V1() V4( NAT ) V5( REAL ) Function-like V14( NAT ) quasi_total V35() V36() V37() Element of Funcs (NAT,REAL) ex b₄ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT st
( b₂ in f & b₃ in Big_Oh (1) & ( for b₅ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT holds
( not b₄ <= b₅ or b₁ . b₅ = (b₂ . b₅) to_power (b₃ . b₅) ) ) ) } is set
N0 is V25() real ext-real Element of REAL
e is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
t is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(t) is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() eventually-nonnegative eventually-positive eventually-nonzero Element of K19(K20(NAT,REAL))
e is V25() real ext-real Element of REAL
f is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
N0 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(N0) is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() eventually-nonnegative eventually-positive eventually-nonzero Element of K19(K20(NAT,REAL))
max (f,2) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
d is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
((- 1)) . d is V25() real ext-real Element of REAL
d to_power (- 1) is V25() real ext-real Element of REAL
f is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
((- 1)) . f is V25() real ext-real Element of REAL
(1) . f is V25() real ext-real Element of REAL
c is V25() real ext-real non negative Element of REAL
f is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() eventually-nonnegative Element of K19(K20(NAT,REAL))
x is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() eventually-nonnegative Element of K19(K20(NAT,REAL))
c + f is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() eventually-nonnegative Element of K19(K20(NAT,REAL))
Big_Oh (c + f) is functional non empty FUNCTION_DOMAIN of NAT , REAL
{ b₁ where b₁ is V1() V4( NAT ) V5( REAL ) Function-like V14( NAT ) quasi_total V35() V36() V37() Element of Funcs (NAT,REAL) : ex b₂ being V25() real ext-real Element of REAL ex b₃ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT st
( not b₂ <= 0 & ( for b₄ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT holds
( not b₃ <= b₄ or ( b₁ . b₄ <= b₂ * ((c + f) . b₄) & 0 <= b₁ . b₄ ) ) ) ) } is set
Big_Oh f is functional non empty FUNCTION_DOMAIN of NAT , REAL
{ b₁ where b₁ is V1() V4( NAT ) V5( REAL ) Function-like V14( NAT ) quasi_total V35() V36() V37() Element of Funcs (NAT,REAL) : ex b₂ being V25() real ext-real Element of REAL ex b₃ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT st
( not b₂ <= 0 & ( for b₄ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT holds
( not b₃ <= b₄ or ( b₁ . b₄ <= b₂ * (f . b₄) & 0 <= b₁ . b₄ ) ) ) ) } is set
e is V25() real ext-real Element of REAL
N0 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
t is V1() V4( NAT ) V5( REAL ) Function-like V14( NAT ) quasi_total V35() V36() V37() Element of Funcs (NAT,REAL)
d is V25() real ext-real Element of REAL
N1 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
d is V25() real ext-real Element of REAL
N1 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 * d is V25() real ext-real Element of REAL
(2 * d) * c is V25() real ext-real Element of REAL
((2 * d) * c) / e is V25() real ext-real Element of REAL
e " is V25() real ext-real set
((2 * d) * c) * (e ") is V25() real ext-real set
max ((2 * d),(((2 * d) * c) / e)) is V25() real ext-real Element of REAL
2 * 0 is 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
max (N0,N1) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
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
t . n is V25() real ext-real Element of REAL
(c + f) . n is V25() real ext-real Element of REAL
d * ((c + f) . n) is V25() real ext-real Element of REAL
f . n is V25() real ext-real Element of REAL
c + (f . n) is V25() real ext-real Element of REAL
d * (c + (f . n)) is V25() real ext-real Element of REAL
(max ((2 * d),(((2 * d) * c) / e))) * (f . n) is V25() real ext-real Element of REAL
d * (f . n) is V25() real ext-real Element of REAL
d * c is V25() real ext-real Element of REAL
(d * c) + (d * (f . n)) is V25() real ext-real Element of REAL
(d * c) + (d * c) is V25() real ext-real Element of REAL
2 * (d * c) is V25() real ext-real Element of REAL
(2 * (d * c)) * 1 is V25() real ext-real Element of REAL
1 / e is V25() real ext-real Element of REAL
1 * (e ") is V25() real ext-real set
(1 / e) * e is V25() real ext-real Element of REAL
(2 * (d * c)) * ((1 / e) * e) is V25() real ext-real Element of REAL
(((2 * d) * c) / e) * e is V25() real ext-real Element of REAL
(max ((2 * d),(((2 * d) * c) / e))) * e is V25() real ext-real Element of REAL
d * (f . n) is V25() real ext-real Element of REAL
d * c is V25() real ext-real Element of REAL
(d * (f . n)) + (d * (f . n)) is V25() real ext-real Element of REAL
(d * c) + (d * (f . n)) is V25() real ext-real Element of REAL
2 * (d * (f . n)) is V25() real ext-real Element of REAL
(2 * d) * (f . n) is V25() real ext-real Element of REAL
c is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
c ^2 is V25() real ext-real Element of REAL
c * c is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer set
2 * c is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(2 * c) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
c is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
c - 1 is V25() real ext-real integer Element of REAL
c + (- 1) is V25() real ext-real integer set
2 to_power (c - 1) is V25() real ext-real Element of REAL
2 * c is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(2 * c) ^2 is V25() real ext-real Element of REAL
(2 * c) * (2 * c) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer set
c is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
c to_power 6 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
c |^ 6 is set
2 * (c to_power 6) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
c + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(c + 1) to_power 6 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(c + 1) |^ 6 is set
c is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 to_power c is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 |^ c is set
c to_power 6 is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
c |^ 6 is set
c is V25() real ext-real Element of REAL
2 to_power c is V25() real ext-real Element of REAL
2 * c is V25() real ext-real Element of REAL
(2 * c) ^2 is V25() real ext-real Element of REAL
(2 * c) * (2 * c) is V25() real ext-real set
c is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
c is V25() real ext-real Element of REAL
f is V25() real ext-real Element of REAL
x is V25() real ext-real Element of REAL
c to_power x is V25() real ext-real Element of REAL
log (f,c) is V25() real ext-real Element of REAL
x * (log (f,c)) is V25() real ext-real Element of REAL
f to_power (x * (log (f,c))) is V25() real ext-real Element of REAL
c is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
c ^2 is V25() real ext-real Element of REAL
c * c is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer set
(c ^2) - c is V25() real ext-real Element of REAL
- c is V25() real ext-real non positive integer set
(c ^2) + (- c) is V25() real ext-real set
((c ^2) - c) + 1 is V25() real ext-real Element of REAL
c is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
c ! is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
x is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
c is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
x ! is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
c ! is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
c is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
c is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
c / 2 is V25() real ext-real non negative Element of REAL
c * (2 ") is V25() real ext-real non negative set
[/(c / 2)\] is V25() real ext-real integer set
c is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
c ! is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
c is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 to_power c is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 |^ c is set
c + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
c is V25() real ext-real logbase Element of REAL
x is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() Element of K19(K20(NAT,REAL))
x . 0 is V25() real ext-real Element of REAL
c is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() eventually-nonnegative Element of K19(K20(NAT,REAL))
x is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() eventually-nonnegative Element of K19(K20(NAT,REAL))
Big_Oh x is functional non empty FUNCTION_DOMAIN of NAT , REAL
{ b₁ where b₁ is V1() V4( NAT ) V5( REAL ) Function-like V14( NAT ) quasi_total V35() V36() V37() Element of Funcs (NAT,REAL) : ex b₂ being V25() real ext-real Element of REAL ex b₃ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT st
( not b₂ <= 0 & ( for b₄ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT holds
( not b₃ <= b₄ or ( b₁ . b₄ <= b₂ * (x . b₄) & 0 <= b₁ . b₄ ) ) ) ) } is set
Big_Oh c is functional non empty FUNCTION_DOMAIN of NAT , REAL
{ b₁ where b₁ is V1() V4( NAT ) V5( REAL ) Function-like V14( NAT ) quasi_total V35() V36() V37() Element of Funcs (NAT,REAL) : ex b₂ being V25() real ext-real Element of REAL ex b₃ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT st
( not b₂ <= 0 & ( for b₄ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT holds
( not b₃ <= b₄ or ( b₁ . b₄ <= b₂ * (c . b₄) & 0 <= b₁ . b₄ ) ) ) ) } is set
f is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() eventually-nonnegative Element of K19(K20(NAT,REAL))
Big_Oh f is functional non empty FUNCTION_DOMAIN of NAT , REAL
{ b₁ where b₁ is V1() V4( NAT ) V5( REAL ) Function-like V14( NAT ) quasi_total V35() V36() V37() Element of Funcs (NAT,REAL) : ex b₂ being V25() real ext-real Element of REAL ex b₃ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT st
( not b₂ <= 0 & ( for b₄ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT holds
( not b₃ <= b₄ or ( b₁ . b₄ <= b₂ * (f . b₄) & 0 <= b₁ . b₄ ) ) ) ) } is set
e is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() eventually-nonnegative Element of K19(K20(NAT,REAL))
Big_Oh e is functional non empty FUNCTION_DOMAIN of NAT , REAL
{ b₁ where b₁ is V1() V4( NAT ) V5( REAL ) Function-like V14( NAT ) quasi_total V35() V36() V37() Element of Funcs (NAT,REAL) : ex b₂ being V25() real ext-real Element of REAL ex b₃ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT st
( not b₂ <= 0 & ( for b₄ being epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT holds
( not b₃ <= b₄ or ( b₁ . b₄ <= b₂ * (e . b₄) & 0 <= b₁ . b₄ ) ) ) ) } is set
c is V25() real ext-real Element of REAL
x is V25() real ext-real Element of REAL
f is V25() real ext-real Element of REAL
c to_power f is V25() real ext-real Element of REAL
x to_power f is V25() real ext-real Element of REAL
c is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 to_power c is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 |^ c is set
2 * c is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(2 * c) + 3 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
c is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 to_power c is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 |^ c is set
c + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(c + 1) ^2 is V25() real ext-real Element of REAL
(c + 1) * (c + 1) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer set
c is V25() real ext-real Element of REAL
2 to_power c is V25() real ext-real Element of REAL
c ^2 is V25() real ext-real Element of REAL
c * c is V25() real ext-real set
x is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() Element of K19(K20(NAT,REAL))
x . 0 is V25() real ext-real Element of REAL
c is non empty V25() real ext-real positive non negative Element of REAL
(c) is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() eventually-nonnegative eventually-positive eventually-nonzero Element of K19(K20(NAT,REAL))
x /" (c) is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() Element of K19(K20(NAT,REAL))
(c) " is V1() V4( NAT ) Function-like V14( NAT ) V35() V36() V37() set
x (#) ((c) ") is V1() V4( NAT ) Function-like V14( NAT ) V35() V36() V37() set
lim (x /" (c)) is V25() real ext-real Element of REAL
c is V25() real ext-real Element of REAL
(c) is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() eventually-nonnegative eventually-positive eventually-nonzero Element of K19(K20(NAT,REAL))
() /" (c) is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() Element of K19(K20(NAT,REAL))
(c) " is V1() V4( NAT ) Function-like V14( NAT ) V35() V36() V37() set
() (#) ((c) ") is V1() V4( NAT ) Function-like V14( NAT ) V35() V36() V37() set
lim (() /" (c)) is V25() real ext-real Element of REAL
x is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
c is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() Element of K19(K20(NAT,REAL))
Sum (c,x) is V25() real ext-real Element of REAL
f is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
c is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() Element of K19(K20(NAT,REAL))
x is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() Element of K19(K20(NAT,REAL))
Sum (c,f) is V25() real ext-real Element of REAL
Sum (x,f) is V25() real ext-real Element of REAL
c is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() Element of K19(K20(NAT,REAL))
c . 0 is V25() real ext-real Element of REAL
x is V25() real ext-real Element of REAL
f is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
Sum (c,f) is V25() real ext-real Element of REAL
x * f is V25() real ext-real Element of REAL
c is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() Element of K19(K20(NAT,REAL))
x is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
f is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
Sum (c,x,f) is V25() real ext-real Element of REAL
x + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
c . (x + 1) is V25() real ext-real Element of REAL
(Sum (c,x,f)) + (c . (x + 1)) is V25() real ext-real Element of REAL
Sum (c,(x + 1),f) is V25() real ext-real Element of REAL
f is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
f + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
e is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
c is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() Element of K19(K20(NAT,REAL))
x is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() Element of K19(K20(NAT,REAL))
Sum (c,e,f) is V25() real ext-real Element of REAL
Sum (x,e,f) is V25() real ext-real Element of REAL
c is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
c / 2 is V25() real ext-real non negative Element of REAL
c * (2 ") is V25() real ext-real non negative set
[/(c / 2)\] is V25() real ext-real integer set
c is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() Element of K19(K20(NAT,REAL))
c . 0 is V25() real ext-real Element of REAL
x is V25() real ext-real Element of REAL
f is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
e is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
Sum (c,f,e) is V25() real ext-real Element of REAL
f - e is V25() real ext-real integer Element of REAL
- e is V25() real ext-real non positive integer set
f + (- e) is V25() real ext-real integer set
x * (f - e) is V25() real ext-real Element of REAL
c is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() Element of K19(K20(NAT,REAL))
lim c is V25() real ext-real Element of REAL
e is V25() real ext-real Element of REAL
f is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
x is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() Element of K19(K20(NAT,REAL))
lim x is V25() real ext-real Element of REAL
c is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
c ^2 is V25() real ext-real Element of REAL
c * c is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer set
(c ^2) - c is V25() real ext-real Element of REAL
- c is V25() real ext-real non positive integer set
(c ^2) + (- c) is V25() real ext-real set
((c ^2) - c) + 1 is V25() real ext-real Element of REAL
c is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
c ^2 is V25() real ext-real Element of REAL
c * c is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer set
(c ^2) - c is V25() real ext-real Element of REAL
- c is V25() real ext-real non positive integer set
(c ^2) + (- c) is V25() real ext-real set
((c ^2) - c) + 1 is V25() real ext-real Element of REAL
2 * (((c ^2) - c) + 1) is V25() real ext-real Element of REAL
c is V25() real ext-real Element of REAL
1 + c is V25() real ext-real Element of REAL
log (2,(1 + c)) is V25() real ext-real Element of REAL
c is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
c - 9 is V25() real ext-real integer Element of REAL
- 9 is V25() real ext-real non positive integer set
c + (- 9) is V25() real ext-real integer set
2 to_power (c - 9) is V25() real ext-real Element of REAL
1 / (2 to_power (c - 9)) is V25() real ext-real Element of REAL
(2 to_power (c - 9)) " is V25() real ext-real set
1 * ((2 to_power (c - 9)) ") is V25() real ext-real set
2 * c is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 to_power (2 * c) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 |^ (2 * c) is set
c ! is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(2 to_power (2 * c)) / (c !) is V25() real ext-real non negative Element of REAL
(c !) " is V25() real ext-real non negative set
(2 to_power (2 * c)) * ((c !) ") is V25() real ext-real non negative set
c is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
c - 1 is V25() real ext-real integer Element of REAL
c + (- 1) is V25() real ext-real integer set
c - 2 is V25() real ext-real integer Element of REAL
- 2 is V25() real ext-real non positive integer set
c + (- 2) is V25() real ext-real integer set
2 * (c - 2) is V25() real ext-real integer Element of REAL
c is V25() real ext-real set
c to_power (1 / 2) is V25() real ext-real set
sqrt c is V25() real ext-real set
c is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
c is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() Element of K19(K20(NAT,REAL))
c is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
c + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(c + 1) / c is V25() real ext-real non negative Element of REAL
c " is V25() real ext-real non negative set
(c + 1) * (c ") is V25() real ext-real non negative set
((c + 1) / c) to_power c is V25() real ext-real Element of REAL
((c + 1) / c) |^ c is set
c + 2 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(c + 2) / (c + 1) is V25() real ext-real non negative Element of REAL
(c + 1) " is non empty V25() real ext-real positive non negative set
(c + 2) * ((c + 1) ") is V25() real ext-real non negative set
((c + 2) / (c + 1)) to_power (c + 1) is V25() real ext-real Element of REAL
((c + 2) / (c + 1)) |^ (c + 1) is set
c is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
x is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
x + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(x + 1) choose c is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
((x + 1) choose c) / (x + 1) is V25() real ext-real non negative Element of REAL
(x + 1) " is non empty V25() real ext-real positive non negative set
((x + 1) choose c) * ((x + 1) ") is V25() real ext-real non negative set
x choose c is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
c is V1() V4( NAT ) V5( REAL ) Function-like non empty V14( NAT ) quasi_total V35() V36() V37() Element of K19(K20(NAT,REAL))
x is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
c . x is V25() real ext-real Element of REAL
Sum ((),x) is V25() real ext-real Element of REAL
c is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 * c is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
log (2,c) is V25() real ext-real Element of REAL
c * (log (2,c)) is V25() real ext-real Element of REAL
c is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
c ^2 is V25() real ext-real Element of REAL
c * c is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer set
c + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
c is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
c + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 to_power (c + 1) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 |^ (c + 1) is set
2 to_power c is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 |^ c is set
(2 to_power (c + 1)) - (2 to_power c) is V25() real ext-real integer Element of REAL
- (2 to_power c) is V25() real ext-real non positive integer set
(2 to_power (c + 1)) + (- (2 to_power c)) is V25() real ext-real integer set
c is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 to_power c is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
2 |^ c is set
(2 to_power c) - 1 is V25() real ext-real integer Element of REAL
(2 to_power c) + (- 1) is V25() real ext-real integer set
x is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
c is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
c ! is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
c + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V25() real ext-real positive non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(c + 1) ! is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
(x) is epsilon-transitive epsilon-connected ordinal natural V25() real ext-real non negative integer V48() V49() V50() V51() V52() V53() V54() Element of NAT
c is V25() real ext-real Element of REAL
x is V25() real ext-real Element of REAL
f is V25() real ext-real Element of REAL
log (x,f) is V25() real ext-real Element of REAL
log (c,f) is V25() real ext-real Element of REAL