:: FDIFF_6 semantic presentation

REAL is non empty V49() V50() V51() V55() V70() set
NAT is non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() Element of K19(REAL)
K19(REAL) is set
COMPLEX is non empty V49() V55() V70() set
{} is empty V15() V16() V17() V19() V20() V21() V23() ext-real non positive non negative V49() V50() V51() V52() V53() V54() V55() V68() set
1 is non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() Element of NAT
{{},1} is non empty V49() V50() V51() V52() V53() V54() set
INT is non empty V49() V50() V51() V52() V53() V55() V70() set
K20(REAL,REAL) is V39() V40() V41() set
K19(K20(REAL,REAL)) is set
K20(NAT,REAL) is V39() V40() V41() set
K19(K20(NAT,REAL)) is set
K20(NAT,COMPLEX) is V39() set
K19(K20(NAT,COMPLEX)) is set
K20(COMPLEX,COMPLEX) is V39() set
K19(K20(COMPLEX,COMPLEX)) is set
PFuncs (REAL,REAL) is set
K20(NAT,(PFuncs (REAL,REAL))) is set
K19(K20(NAT,(PFuncs (REAL,REAL)))) is set
RAT is non empty V49() V50() V51() V52() V55() V70() set
NAT is non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() set
K19(NAT) is set
K19(NAT) is set
0 is empty V15() V16() V17() V19() V20() V21() V22() V23() ext-real non positive non negative V49() V50() V51() V52() V53() V54() V55() V68() V69() Element of NAT
number_e is V22() V23() ext-real Element of REAL
2 is non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() Element of NAT
exp_R is Relation-like REAL -defined REAL -valued V6() V30( REAL , REAL ) V39() V40() V41() Element of K19(K20(REAL,REAL))
[#] REAL is V49() V50() V51() Element of K19(REAL)
K326(REAL,exp_R) is V49() V50() V51() Element of K19(REAL)
K327(REAL,exp_R) is V49() V50() V51() Element of K19(REAL)
right_open_halfline 0 is V49() V50() V51() Element of K19(REAL)
ln is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
log_ number_e is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
K162(REAL,REAL,exp_R) is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
K326(REAL,ln) is V49() V50() V51() Element of K19(REAL)
K327(REAL,ln) is V49() V50() V51() Element of K19(REAL)
{0} is non empty V49() V50() V51() V52() V53() V54() set
sin is Relation-like REAL -defined REAL -valued V6() V30( REAL , REAL ) V39() V40() V41() Element of K19(K20(REAL,REAL))
cos is Relation-like REAL -defined REAL -valued V6() V30( REAL , REAL ) V39() V40() V41() Element of K19(K20(REAL,REAL))
exp_R . 0 is V22() V23() ext-real Element of REAL
#Z 2 is Relation-like REAL -defined REAL -valued V6() V30( REAL , REAL ) V39() V40() V41() Element of K19(K20(REAL,REAL))
Z is V22() V23() ext-real Element of REAL
2 * Z is V22() V23() ext-real Element of REAL
cos (2 * Z) is V22() V23() ext-real Element of REAL
1 - (cos (2 * Z)) is V22() V23() ext-real Element of REAL
sin Z is V22() V23() ext-real Element of REAL
(sin Z) ^2 is V22() V23() ext-real Element of REAL
K57((sin Z),(sin Z)) is set
2 * ((sin Z) ^2) is V22() V23() ext-real Element of REAL
Z + Z is V22() V23() ext-real Element of REAL
cos (Z + Z) is V22() V23() ext-real Element of REAL
1 - (cos (Z + Z)) is V22() V23() ext-real Element of REAL
cos Z is V22() V23() ext-real Element of REAL
(cos Z) * (cos Z) is V22() V23() ext-real Element of REAL
(sin Z) * (sin Z) is V22() V23() ext-real Element of REAL
((cos Z) * (cos Z)) - ((sin Z) * (sin Z)) is V22() V23() ext-real Element of REAL
1 - (((cos Z) * (cos Z)) - ((sin Z) * (sin Z))) is V22() V23() ext-real Element of REAL
(cos Z) ^2 is V22() V23() ext-real Element of REAL
K57((cos Z),(cos Z)) is set
((cos Z) ^2) + ((sin Z) ^2) is V22() V23() ext-real Element of REAL
((cos Z) ^2) - ((sin Z) ^2) is V22() V23() ext-real Element of REAL
(((cos Z) ^2) + ((sin Z) ^2)) - (((cos Z) ^2) - ((sin Z) ^2)) is V22() V23() ext-real Element of REAL
Z is V22() V23() ext-real Element of REAL
2 * Z is V22() V23() ext-real Element of REAL
cos (2 * Z) is V22() V23() ext-real Element of REAL
1 + (cos (2 * Z)) is V22() V23() ext-real Element of REAL
cos Z is V22() V23() ext-real Element of REAL
(cos Z) ^2 is V22() V23() ext-real Element of REAL
K57((cos Z),(cos Z)) is set
2 * ((cos Z) ^2) is V22() V23() ext-real Element of REAL
Z + Z is V22() V23() ext-real Element of REAL
cos (Z + Z) is V22() V23() ext-real Element of REAL
1 + (cos (Z + Z)) is V22() V23() ext-real Element of REAL
sin Z is V22() V23() ext-real Element of REAL
(sin Z) ^2 is V22() V23() ext-real Element of REAL
K57((sin Z),(sin Z)) is set
((cos Z) ^2) - ((sin Z) ^2) is V22() V23() ext-real Element of REAL
1 + (((cos Z) ^2) - ((sin Z) ^2)) is V22() V23() ext-real Element of REAL
((cos Z) ^2) + ((sin Z) ^2) is V22() V23() ext-real Element of REAL
(((cos Z) ^2) + ((sin Z) ^2)) + (((cos Z) ^2) - ((sin Z) ^2)) is V22() V23() ext-real Element of REAL
1 / 2 is V22() V23() ext-real Element of REAL
Z is V22() V23() ext-real Element of REAL
sin . Z is V22() V23() ext-real Element of REAL
cos . Z is V22() V23() ext-real Element of REAL
1 - (cos . Z) is V22() V23() ext-real Element of REAL
1 + (cos . Z) is V22() V23() ext-real Element of REAL
(1 - (cos . Z)) * (1 + (cos . Z)) is V22() V23() ext-real Element of REAL
((1 - (cos . Z)) * (1 + (cos . Z))) #R (1 / 2) is V22() V23() ext-real Element of REAL
2 * (1 / 2) is V22() V23() ext-real Element of REAL
(sin . Z) #R (2 * (1 / 2)) is V22() V23() ext-real Element of REAL
1 + 1 is V15() V16() V17() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() Element of NAT
(sin . Z) #R (1 + 1) is V22() V23() ext-real Element of REAL
((sin . Z) #R (1 + 1)) #R (1 / 2) is V22() V23() ext-real Element of REAL
(sin . Z) #R 1 is V22() V23() ext-real Element of REAL
((sin . Z) #R 1) * ((sin . Z) #R 1) is V22() V23() ext-real Element of REAL
(((sin . Z) #R 1) * ((sin . Z) #R 1)) #R (1 / 2) is V22() V23() ext-real Element of REAL
(sin . Z) * ((sin . Z) #R 1) is V22() V23() ext-real Element of REAL
((sin . Z) * ((sin . Z) #R 1)) #R (1 / 2) is V22() V23() ext-real Element of REAL
(sin . Z) ^2 is V22() V23() ext-real Element of REAL
K57((sin . Z),(sin . Z)) is set
(cos . Z) ^2 is V22() V23() ext-real Element of REAL
K57((cos . Z),(cos . Z)) is set
((sin . Z) ^2) + ((cos . Z) ^2) is V22() V23() ext-real Element of REAL
(((sin . Z) ^2) + ((cos . Z) ^2)) - ((cos . Z) ^2) is V22() V23() ext-real Element of REAL
((((sin . Z) ^2) + ((cos . Z) ^2)) - ((cos . Z) ^2)) #R (1 / 2) is V22() V23() ext-real Element of REAL
1 ^2 is V22() V23() ext-real Element of REAL
K57(1,1) is V23() V68() set
(1 ^2) - ((cos . Z) ^2) is V22() V23() ext-real Element of REAL
((1 ^2) - ((cos . Z) ^2)) #R (1 / 2) is V22() V23() ext-real Element of REAL
- 1 is V22() V23() ext-real V68() Element of REAL
Z is V22() V23() ext-real Element of REAL
sin . Z is V22() V23() ext-real Element of REAL
cos . Z is V22() V23() ext-real Element of REAL
1 - (cos . Z) is V22() V23() ext-real Element of REAL
(1 - (cos . Z)) #R (1 / 2) is V22() V23() ext-real Element of REAL
(sin . Z) / ((1 - (cos . Z)) #R (1 / 2)) is V22() V23() ext-real Element of REAL
1 + (cos . Z) is V22() V23() ext-real Element of REAL
(1 + (cos . Z)) #R (1 / 2) is V22() V23() ext-real Element of REAL
1 - 1 is V22() V23() ext-real V68() Element of REAL
1 + (- 1) is V22() V23() ext-real V68() Element of REAL
(1 - (cos . Z)) * (1 + (cos . Z)) is V22() V23() ext-real Element of REAL
((1 - (cos . Z)) * (1 + (cos . Z))) #R (1 / 2) is V22() V23() ext-real Element of REAL
(((1 - (cos . Z)) * (1 + (cos . Z))) #R (1 / 2)) / ((1 - (cos . Z)) #R (1 / 2)) is V22() V23() ext-real Element of REAL
((1 - (cos . Z)) * (1 + (cos . Z))) / (1 - (cos . Z)) is V22() V23() ext-real Element of REAL
(((1 - (cos . Z)) * (1 + (cos . Z))) / (1 - (cos . Z))) #R (1 / 2) is V22() V23() ext-real Element of REAL
Z is V22() V23() ext-real Element of REAL
log (number_e,Z) is V22() V23() ext-real Element of REAL
x is V22() V23() ext-real Element of REAL
x * (log (number_e,Z)) is V22() V23() ext-real Element of REAL
exp_R . (x * (log (number_e,Z))) is V22() V23() ext-real Element of REAL
Z #R x is V22() V23() ext-real Element of REAL
Z to_power x is V22() V23() ext-real Element of REAL
log (number_e,(Z to_power x)) is V22() V23() ext-real Element of REAL
exp_R . (log (number_e,(Z to_power x))) is V22() V23() ext-real Element of REAL
log (number_e,(Z #R x)) is V22() V23() ext-real Element of REAL
exp_R . (log (number_e,(Z #R x))) is V22() V23() ext-real Element of REAL
Z is V22() V23() ext-real Element of REAL
log (number_e,Z) is V22() V23() ext-real Element of REAL
x is V22() V23() ext-real Element of REAL
x * (log (number_e,Z)) is V22() V23() ext-real Element of REAL
- (x * (log (number_e,Z))) is V22() V23() ext-real Element of REAL
exp_R . (- (x * (log (number_e,Z)))) is V22() V23() ext-real Element of REAL
- x is V22() V23() ext-real Element of REAL
Z #R (- x) is V22() V23() ext-real Element of REAL
(- x) * (log (number_e,Z)) is V22() V23() ext-real Element of REAL
exp_R . ((- x) * (log (number_e,Z))) is V22() V23() ext-real Element of REAL
Z to_power (- x) is V22() V23() ext-real Element of REAL
log (number_e,(Z to_power (- x))) is V22() V23() ext-real Element of REAL
exp_R . (log (number_e,(Z to_power (- x)))) is V22() V23() ext-real Element of REAL
log (number_e,(Z #R (- x))) is V22() V23() ext-real Element of REAL
exp_R . (log (number_e,(Z #R (- x)))) is V22() V23() ext-real Element of REAL
Z is V22() V23() ext-real Element of REAL
sin . Z is V22() V23() ext-real Element of REAL
cos . Z is V22() V23() ext-real Element of REAL
1 + (cos . Z) is V22() V23() ext-real Element of REAL
(1 + (cos . Z)) #R (1 / 2) is V22() V23() ext-real Element of REAL
(sin . Z) / ((1 + (cos . Z)) #R (1 / 2)) is V22() V23() ext-real Element of REAL
1 - (cos . Z) is V22() V23() ext-real Element of REAL
(1 - (cos . Z)) #R (1 / 2) is V22() V23() ext-real Element of REAL
1 + (- 1) is V22() V23() ext-real V68() Element of REAL
1 - 1 is V22() V23() ext-real V68() Element of REAL
(1 - (cos . Z)) * (1 + (cos . Z)) is V22() V23() ext-real Element of REAL
((1 - (cos . Z)) * (1 + (cos . Z))) #R (1 / 2) is V22() V23() ext-real Element of REAL
(((1 - (cos . Z)) * (1 + (cos . Z))) #R (1 / 2)) / ((1 + (cos . Z)) #R (1 / 2)) is V22() V23() ext-real Element of REAL
((1 - (cos . Z)) * (1 + (cos . Z))) / (1 + (cos . Z)) is V22() V23() ext-real Element of REAL
(((1 - (cos . Z)) * (1 + (cos . Z))) / (1 + (cos . Z))) #R (1 / 2) is V22() V23() ext-real Element of REAL
Z is V22() V23() ext-real Element of REAL
Z ^2 is V22() V23() ext-real Element of REAL
K57(Z,Z) is set
x is open V49() V50() V51() Element of K19(REAL)
x is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
f is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
x - f is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
- f is Relation-like V6() V39() set
K58(1) is V22() V23() V68() set
K58(1) (#) f is Relation-like V6() set
x + (- f) is Relation-like V6() set
dom (x - f) is set
(x - f) `| x is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
(- 1) (#) f is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
x + ((- 1) (#) f) is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
x is V22() V23() ext-real Element of REAL
x . x is V22() V23() ext-real Element of REAL
0 * x is V22() V23() ext-real Element of REAL
(Z ^2) + (0 * x) is V22() V23() ext-real Element of REAL
x is V22() V23() ext-real Element of REAL
((x - f) `| x) . x is V22() V23() ext-real Element of REAL
2 * x is V22() V23() ext-real Element of REAL
- (2 * x) is V22() V23() ext-real Element of REAL
2 * (- 1) is V22() V23() ext-real V68() Element of REAL
(2 * (- 1)) * x is V22() V23() ext-real Element of REAL
0 + ((2 * (- 1)) * x) is V22() V23() ext-real Element of REAL
x is V22() V23() ext-real Element of REAL
((x - f) `| x) . x is V22() V23() ext-real Element of REAL
2 * x is V22() V23() ext-real Element of REAL
- (2 * x) is V22() V23() ext-real Element of REAL
4 is non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() Element of NAT
Z is V22() V23() ext-real Element of REAL
Z ^2 is V22() V23() ext-real Element of REAL
K57(Z,Z) is set
4 * (Z ^2) is V22() V23() ext-real Element of REAL
x is open V49() V50() V51() Element of K19(REAL)
x is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
f is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
x + f is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
x - f is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
- f is Relation-like V6() V39() set
K58(1) is V22() V23() V68() set
K58(1) (#) f is Relation-like V6() set
x + (- f) is Relation-like V6() set
(x + f) / (x - f) is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
dom ((x + f) / (x - f)) is set
((x + f) / (x - f)) `| x is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
x is V22() V23() ext-real Element of REAL
x . x is V22() V23() ext-real Element of REAL
dom (x - f) is set
dom (x + f) is set
(x - f) " {0} is set
(dom (x - f)) \ ((x - f) " {0}) is Element of K19((dom (x - f)))
K19((dom (x - f))) is set
(dom (x + f)) /\ ((dom (x - f)) \ ((x - f) " {0})) is Element of K19((dom (x - f)))
x is V22() V23() ext-real Element of REAL
(x - f) . x is V22() V23() ext-real Element of REAL
x is V22() V23() ext-real Element of REAL
(((x + f) / (x - f)) `| x) . x is V22() V23() ext-real Element of REAL
(4 * (Z ^2)) * x is V22() V23() ext-real Element of REAL
x |^ 2 is V22() V23() ext-real Element of REAL
(Z ^2) - (x |^ 2) is V22() V23() ext-real Element of REAL
((Z ^2) - (x |^ 2)) ^2 is V22() V23() ext-real Element of REAL
K57(((Z ^2) - (x |^ 2)),((Z ^2) - (x |^ 2))) is set
((4 * (Z ^2)) * x) / (((Z ^2) - (x |^ 2)) ^2) is V22() V23() ext-real Element of REAL
f . x is V22() V23() ext-real Element of REAL
x #Z 2 is V22() V23() ext-real Element of REAL
(x - f) . x is V22() V23() ext-real Element of REAL
x . x is V22() V23() ext-real Element of REAL
(x . x) - (f . x) is V22() V23() ext-real Element of REAL
(x + f) . x is V22() V23() ext-real Element of REAL
(x . x) + (f . x) is V22() V23() ext-real Element of REAL
(Z ^2) + (x |^ 2) is V22() V23() ext-real Element of REAL
diff (((x + f) / (x - f)),x) is V22() V23() ext-real Element of REAL
diff ((x + f),x) is V22() V23() ext-real Element of REAL
(diff ((x + f),x)) * ((x - f) . x) is V22() V23() ext-real Element of REAL
diff ((x - f),x) is V22() V23() ext-real Element of REAL
(diff ((x - f),x)) * ((x + f) . x) is V22() V23() ext-real Element of REAL
((diff ((x + f),x)) * ((x - f) . x)) - ((diff ((x - f),x)) * ((x + f) . x)) is V22() V23() ext-real Element of REAL
((x - f) . x) ^2 is V22() V23() ext-real Element of REAL
K57(((x - f) . x),((x - f) . x)) is set
(((diff ((x + f),x)) * ((x - f) . x)) - ((diff ((x - f),x)) * ((x + f) . x))) / (((x - f) . x) ^2) is V22() V23() ext-real Element of REAL
(x + f) `| x is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
((x + f) `| x) . x is V22() V23() ext-real Element of REAL
(((x + f) `| x) . x) * ((x - f) . x) is V22() V23() ext-real Element of REAL
((((x + f) `| x) . x) * ((x - f) . x)) - ((diff ((x - f),x)) * ((x + f) . x)) is V22() V23() ext-real Element of REAL
(((((x + f) `| x) . x) * ((x - f) . x)) - ((diff ((x - f),x)) * ((x + f) . x))) / (((x - f) . x) ^2) is V22() V23() ext-real Element of REAL
(x - f) `| x is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
((x - f) `| x) . x is V22() V23() ext-real Element of REAL
(((x - f) `| x) . x) * ((x + f) . x) is V22() V23() ext-real Element of REAL
((((x + f) `| x) . x) * ((x - f) . x)) - ((((x - f) `| x) . x) * ((x + f) . x)) is V22() V23() ext-real Element of REAL
(((((x + f) `| x) . x) * ((x - f) . x)) - ((((x - f) `| x) . x) * ((x + f) . x))) / (((x - f) . x) ^2) is V22() V23() ext-real Element of REAL
2 * x is V22() V23() ext-real Element of REAL
(2 * x) * ((x - f) . x) is V22() V23() ext-real Element of REAL
((2 * x) * ((x - f) . x)) - ((((x - f) `| x) . x) * ((x + f) . x)) is V22() V23() ext-real Element of REAL
(((2 * x) * ((x - f) . x)) - ((((x - f) `| x) . x) * ((x + f) . x))) / (((x - f) . x) ^2) is V22() V23() ext-real Element of REAL
- (2 * x) is V22() V23() ext-real Element of REAL
(- (2 * x)) * ((x + f) . x) is V22() V23() ext-real Element of REAL
((2 * x) * ((x - f) . x)) - ((- (2 * x)) * ((x + f) . x)) is V22() V23() ext-real Element of REAL
(((2 * x) * ((x - f) . x)) - ((- (2 * x)) * ((x + f) . x))) / (((x - f) . x) ^2) is V22() V23() ext-real Element of REAL
x is V22() V23() ext-real Element of REAL
(((x + f) / (x - f)) `| x) . x is V22() V23() ext-real Element of REAL
(4 * (Z ^2)) * x is V22() V23() ext-real Element of REAL
x |^ 2 is V22() V23() ext-real Element of REAL
(Z ^2) - (x |^ 2) is V22() V23() ext-real Element of REAL
((Z ^2) - (x |^ 2)) ^2 is V22() V23() ext-real Element of REAL
K57(((Z ^2) - (x |^ 2)),((Z ^2) - (x |^ 2))) is set
((4 * (Z ^2)) * x) / (((Z ^2) - (x |^ 2)) ^2) is V22() V23() ext-real Element of REAL
Z is V22() V23() ext-real Element of REAL
Z ^2 is V22() V23() ext-real Element of REAL
K57(Z,Z) is set
4 * (Z ^2) is V22() V23() ext-real Element of REAL
Z |^ 4 is V22() V23() ext-real Element of REAL
x is open V49() V50() V51() Element of K19(REAL)
x is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
dom x is set
x `| x is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
f is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
x is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
f + x is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
f - x is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
- x is Relation-like V6() V39() set
K58(1) is V22() V23() V68() set
K58(1) (#) x is Relation-like V6() set
f + (- x) is Relation-like V6() set
(f + x) / (f - x) is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
ln * ((f + x) / (f - x)) is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
dom ((f + x) / (f - x)) is set
x is set
dom (f - x) is set
dom (f + x) is set
(f - x) " {0} is set
(dom (f - x)) \ ((f - x) " {0}) is Element of K19((dom (f - x)))
K19((dom (f - x))) is set
(dom (f + x)) /\ ((dom (f - x)) \ ((f - x) " {0})) is Element of K19((dom (f - x)))
x is V22() V23() ext-real Element of REAL
f . x is V22() V23() ext-real Element of REAL
(f - x) . x is V22() V23() ext-real Element of REAL
x is V22() V23() ext-real Element of REAL
((f + x) / (f - x)) . x is V22() V23() ext-real Element of REAL
x . x is V22() V23() ext-real Element of REAL
x #Z 2 is V22() V23() ext-real Element of REAL
x |^ 2 is V22() V23() ext-real Element of REAL
(f - x) . x is V22() V23() ext-real Element of REAL
x ^2 is V22() V23() ext-real Element of REAL
K57(x,x) is set
(Z ^2) + (x |^ 2) is V22() V23() ext-real Element of REAL
0 + 0 is V15() V16() V17() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() Element of NAT
(f + x) . x is V22() V23() ext-real Element of REAL
((f - x) . x) " is V22() V23() ext-real Element of REAL
((f + x) . x) * (((f - x) . x) ") is V22() V23() ext-real Element of REAL
((f + x) . x) / ((f - x) . x) is V22() V23() ext-real Element of REAL
f . x is V22() V23() ext-real Element of REAL
(f . x) + (x . x) is V22() V23() ext-real Element of REAL
x is V22() V23() ext-real Element of REAL
((f + x) / (f - x)) . x is V22() V23() ext-real Element of REAL
x is V22() V23() ext-real Element of REAL
(x `| x) . x is V22() V23() ext-real Element of REAL
(4 * (Z ^2)) * x is V22() V23() ext-real Element of REAL
x |^ 4 is V22() V23() ext-real Element of REAL
(Z |^ 4) - (x |^ 4) is V22() V23() ext-real Element of REAL
((4 * (Z ^2)) * x) / ((Z |^ 4) - (x |^ 4)) is V22() V23() ext-real Element of REAL
(Z ^2) ^2 is V22() V23() ext-real Element of REAL
K57((Z ^2),(Z ^2)) is set
Z |^ 2 is V22() V23() ext-real Element of REAL
(Z |^ 2) * (Z ^2) is V22() V23() ext-real Element of REAL
(Z |^ 2) * (Z |^ 2) is V22() V23() ext-real Element of REAL
(Z |^ 2) |^ 2 is V22() V23() ext-real Element of REAL
2 * 2 is V15() V16() V17() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() Element of NAT
Z |^ (2 * 2) is V22() V23() ext-real Element of REAL
x |^ 2 is V22() V23() ext-real Element of REAL
(x |^ 2) ^2 is V22() V23() ext-real Element of REAL
K57((x |^ 2),(x |^ 2)) is set
(x |^ 2) |^ 2 is V22() V23() ext-real Element of REAL
x |^ (2 * 2) is V22() V23() ext-real Element of REAL
x . x is V22() V23() ext-real Element of REAL
x #Z 2 is V22() V23() ext-real Element of REAL
(f + x) . x is V22() V23() ext-real Element of REAL
f . x is V22() V23() ext-real Element of REAL
(f . x) + (x . x) is V22() V23() ext-real Element of REAL
(Z ^2) + (x |^ 2) is V22() V23() ext-real Element of REAL
(f - x) . x is V22() V23() ext-real Element of REAL
(f . x) - (x . x) is V22() V23() ext-real Element of REAL
(Z ^2) - (x |^ 2) is V22() V23() ext-real Element of REAL
((f + x) / (f - x)) . x is V22() V23() ext-real Element of REAL
((f - x) . x) " is V22() V23() ext-real Element of REAL
((f + x) . x) * (((f - x) . x) ") is V22() V23() ext-real Element of REAL
((Z ^2) + (x |^ 2)) / ((Z ^2) - (x |^ 2)) is V22() V23() ext-real Element of REAL
diff ((ln * ((f + x) / (f - x))),x) is V22() V23() ext-real Element of REAL
diff (((f + x) / (f - x)),x) is V22() V23() ext-real Element of REAL
(diff (((f + x) / (f - x)),x)) / (((f + x) / (f - x)) . x) is V22() V23() ext-real Element of REAL
((f + x) / (f - x)) `| x is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
(((f + x) / (f - x)) `| x) . x is V22() V23() ext-real Element of REAL
((((f + x) / (f - x)) `| x) . x) / (((f + x) / (f - x)) . x) is V22() V23() ext-real Element of REAL
((Z ^2) - (x |^ 2)) ^2 is V22() V23() ext-real Element of REAL
K57(((Z ^2) - (x |^ 2)),((Z ^2) - (x |^ 2))) is set
((4 * (Z ^2)) * x) / (((Z ^2) - (x |^ 2)) ^2) is V22() V23() ext-real Element of REAL
(((4 * (Z ^2)) * x) / (((Z ^2) - (x |^ 2)) ^2)) / (((Z ^2) + (x |^ 2)) / ((Z ^2) - (x |^ 2))) is V22() V23() ext-real Element of REAL
((4 * (Z ^2)) * x) / ((Z ^2) - (x |^ 2)) is V22() V23() ext-real Element of REAL
(((4 * (Z ^2)) * x) / ((Z ^2) - (x |^ 2))) / ((Z ^2) - (x |^ 2)) is V22() V23() ext-real Element of REAL
((((4 * (Z ^2)) * x) / ((Z ^2) - (x |^ 2))) / ((Z ^2) - (x |^ 2))) / (((Z ^2) + (x |^ 2)) / ((Z ^2) - (x |^ 2))) is V22() V23() ext-real Element of REAL
(((4 * (Z ^2)) * x) / ((Z ^2) - (x |^ 2))) / (((Z ^2) + (x |^ 2)) / ((Z ^2) - (x |^ 2))) is V22() V23() ext-real Element of REAL
((((4 * (Z ^2)) * x) / ((Z ^2) - (x |^ 2))) / (((Z ^2) + (x |^ 2)) / ((Z ^2) - (x |^ 2)))) / ((Z ^2) - (x |^ 2)) is V22() V23() ext-real Element of REAL
((4 * (Z ^2)) * x) / ((Z ^2) + (x |^ 2)) is V22() V23() ext-real Element of REAL
(((4 * (Z ^2)) * x) / ((Z ^2) + (x |^ 2))) / ((Z ^2) - (x |^ 2)) is V22() V23() ext-real Element of REAL
((Z ^2) + (x |^ 2)) * ((Z ^2) - (x |^ 2)) is V22() V23() ext-real Element of REAL
((4 * (Z ^2)) * x) / (((Z ^2) + (x |^ 2)) * ((Z ^2) - (x |^ 2))) is V22() V23() ext-real Element of REAL
x is V22() V23() ext-real Element of REAL
(x `| x) . x is V22() V23() ext-real Element of REAL
(4 * (Z ^2)) * x is V22() V23() ext-real Element of REAL
x |^ 4 is V22() V23() ext-real Element of REAL
(Z |^ 4) - (x |^ 4) is V22() V23() ext-real Element of REAL
((4 * (Z ^2)) * x) / ((Z |^ 4) - (x |^ 4)) is V22() V23() ext-real Element of REAL
Z is V22() V23() ext-real Element of REAL
Z ^2 is V22() V23() ext-real Element of REAL
K57(Z,Z) is set
4 * (Z ^2) is V22() V23() ext-real Element of REAL
1 / (4 * (Z ^2)) is V22() V23() ext-real Element of REAL
Z |^ 4 is V22() V23() ext-real Element of REAL
x is open V49() V50() V51() Element of K19(REAL)
x is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
(1 / (4 * (Z ^2))) (#) x is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
dom ((1 / (4 * (Z ^2))) (#) x) is set
((1 / (4 * (Z ^2))) (#) x) `| x is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
f is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
x is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
f + x is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
f - x is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
- x is Relation-like V6() V39() set
K58(1) is V22() V23() V68() set
K58(1) (#) x is Relation-like V6() set
f + (- x) is Relation-like V6() set
(f + x) / (f - x) is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
ln * ((f + x) / (f - x)) is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
dom x is set
x is V22() V23() ext-real Element of REAL
(((1 / (4 * (Z ^2))) (#) x) `| x) . x is V22() V23() ext-real Element of REAL
x |^ 4 is V22() V23() ext-real Element of REAL
(Z |^ 4) - (x |^ 4) is V22() V23() ext-real Element of REAL
x / ((Z |^ 4) - (x |^ 4)) is V22() V23() ext-real Element of REAL
4 * 0 is V15() V16() V17() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() Element of NAT
diff (x,x) is V22() V23() ext-real Element of REAL
(1 / (4 * (Z ^2))) * (diff (x,x)) is V22() V23() ext-real Element of REAL
x `| x is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
(x `| x) . x is V22() V23() ext-real Element of REAL
(1 / (4 * (Z ^2))) * ((x `| x) . x) is V22() V23() ext-real Element of REAL
(4 * (Z ^2)) * x is V22() V23() ext-real Element of REAL
((4 * (Z ^2)) * x) / ((Z |^ 4) - (x |^ 4)) is V22() V23() ext-real Element of REAL
(1 / (4 * (Z ^2))) * (((4 * (Z ^2)) * x) / ((Z |^ 4) - (x |^ 4))) is V22() V23() ext-real Element of REAL
(4 * (Z ^2)) * (x / ((Z |^ 4) - (x |^ 4))) is V22() V23() ext-real Element of REAL
(1 / (4 * (Z ^2))) * ((4 * (Z ^2)) * (x / ((Z |^ 4) - (x |^ 4)))) is V22() V23() ext-real Element of REAL
(1 / (4 * (Z ^2))) * (4 * (Z ^2)) is V22() V23() ext-real Element of REAL
(x / ((Z |^ 4) - (x |^ 4))) * ((1 / (4 * (Z ^2))) * (4 * (Z ^2))) is V22() V23() ext-real Element of REAL
x is V22() V23() ext-real Element of REAL
(((1 / (4 * (Z ^2))) (#) x) `| x) . x is V22() V23() ext-real Element of REAL
x |^ 4 is V22() V23() ext-real Element of REAL
(Z |^ 4) - (x |^ 4) is V22() V23() ext-real Element of REAL
x / ((Z |^ 4) - (x |^ 4)) is V22() V23() ext-real Element of REAL
Z is open V49() V50() V51() Element of K19(REAL)
x is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
x is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
x + x is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
x / (x + x) is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
dom (x / (x + x)) is set
(x / (x + x)) `| Z is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
dom (x + x) is set
dom x is set
(x + x) " {0} is set
(dom (x + x)) \ ((x + x) " {0}) is Element of K19((dom (x + x)))
K19((dom (x + x))) is set
(dom x) /\ ((dom (x + x)) \ ((x + x) " {0})) is Element of K19((dom (x + x)))
f is V22() V23() ext-real Element of REAL
x `| Z is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
f is V22() V23() ext-real Element of REAL
(x `| Z) . f is V22() V23() ext-real Element of REAL
2 * f is V22() V23() ext-real Element of REAL
2 - 1 is V22() V23() ext-real V68() Element of REAL
f #Z (2 - 1) is V22() V23() ext-real Element of REAL
2 * (f #Z (2 - 1)) is V22() V23() ext-real Element of REAL
diff (x,f) is V22() V23() ext-real Element of REAL
1 ^2 is V22() V23() ext-real Element of REAL
K57(1,1) is V23() V68() set
f is V22() V23() ext-real Element of REAL
x . f is V22() V23() ext-real Element of REAL
f is V22() V23() ext-real Element of REAL
(x + x) . f is V22() V23() ext-real Element of REAL
f ^2 is V22() V23() ext-real Element of REAL
K57(f,f) is set
1 + (f ^2) is V22() V23() ext-real Element of REAL
0 + 0 is V15() V16() V17() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() Element of NAT
x . f is V22() V23() ext-real Element of REAL
x . f is V22() V23() ext-real Element of REAL
(x . f) + (x . f) is V22() V23() ext-real Element of REAL
1 + (x . f) is V22() V23() ext-real Element of REAL
f #Z 2 is V22() V23() ext-real Element of REAL
1 + (f #Z 2) is V22() V23() ext-real Element of REAL
f |^ 2 is V22() V23() ext-real Element of REAL
1 + (f |^ 2) is V22() V23() ext-real Element of REAL
f is V22() V23() ext-real Element of REAL
((x / (x + x)) `| Z) . f is V22() V23() ext-real Element of REAL
2 * f is V22() V23() ext-real Element of REAL
f ^2 is V22() V23() ext-real Element of REAL
K57(f,f) is set
1 + (f ^2) is V22() V23() ext-real Element of REAL
(1 + (f ^2)) ^2 is V22() V23() ext-real Element of REAL
K57((1 + (f ^2)),(1 + (f ^2))) is set
(2 * f) / ((1 + (f ^2)) ^2) is V22() V23() ext-real Element of REAL
x . f is V22() V23() ext-real Element of REAL
f #Z 2 is V22() V23() ext-real Element of REAL
f |^ 2 is V22() V23() ext-real Element of REAL
(x + x) . f is V22() V23() ext-real Element of REAL
x . f is V22() V23() ext-real Element of REAL
(x . f) + (x . f) is V22() V23() ext-real Element of REAL
1 + (x . f) is V22() V23() ext-real Element of REAL
1 + (f #Z 2) is V22() V23() ext-real Element of REAL
1 + (f |^ 2) is V22() V23() ext-real Element of REAL
diff ((x / (x + x)),f) is V22() V23() ext-real Element of REAL
diff (x,f) is V22() V23() ext-real Element of REAL
(diff (x,f)) * ((x + x) . f) is V22() V23() ext-real Element of REAL
diff ((x + x),f) is V22() V23() ext-real Element of REAL
(diff ((x + x),f)) * (x . f) is V22() V23() ext-real Element of REAL
((diff (x,f)) * ((x + x) . f)) - ((diff ((x + x),f)) * (x . f)) is V22() V23() ext-real Element of REAL
((x + x) . f) ^2 is V22() V23() ext-real Element of REAL
K57(((x + x) . f),((x + x) . f)) is set
(((diff (x,f)) * ((x + x) . f)) - ((diff ((x + x),f)) * (x . f))) / (((x + x) . f) ^2) is V22() V23() ext-real Element of REAL
(x `| Z) . f is V22() V23() ext-real Element of REAL
((x `| Z) . f) * ((x + x) . f) is V22() V23() ext-real Element of REAL
(((x `| Z) . f) * ((x + x) . f)) - ((diff ((x + x),f)) * (x . f)) is V22() V23() ext-real Element of REAL
((((x `| Z) . f) * ((x + x) . f)) - ((diff ((x + x),f)) * (x . f))) / (((x + x) . f) ^2) is V22() V23() ext-real Element of REAL
(x + x) `| Z is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
((x + x) `| Z) . f is V22() V23() ext-real Element of REAL
(((x + x) `| Z) . f) * (x . f) is V22() V23() ext-real Element of REAL
(((x `| Z) . f) * ((x + x) . f)) - ((((x + x) `| Z) . f) * (x . f)) is V22() V23() ext-real Element of REAL
((((x `| Z) . f) * ((x + x) . f)) - ((((x + x) `| Z) . f) * (x . f))) / (((x + x) . f) ^2) is V22() V23() ext-real Element of REAL
(2 * f) * ((x + x) . f) is V22() V23() ext-real Element of REAL
((2 * f) * ((x + x) . f)) - ((((x + x) `| Z) . f) * (x . f)) is V22() V23() ext-real Element of REAL
(((2 * f) * ((x + x) . f)) - ((((x + x) `| Z) . f) * (x . f))) / (((x + x) . f) ^2) is V22() V23() ext-real Element of REAL
(2 * f) * (1 + (f ^2)) is V22() V23() ext-real Element of REAL
(2 * f) * (f ^2) is V22() V23() ext-real Element of REAL
((2 * f) * (1 + (f ^2))) - ((2 * f) * (f ^2)) is V22() V23() ext-real Element of REAL
(((2 * f) * (1 + (f ^2))) - ((2 * f) * (f ^2))) / ((1 + (f ^2)) ^2) is V22() V23() ext-real Element of REAL
f is V22() V23() ext-real Element of REAL
((x / (x + x)) `| Z) . f is V22() V23() ext-real Element of REAL
2 * f is V22() V23() ext-real Element of REAL
f ^2 is V22() V23() ext-real Element of REAL
K57(f,f) is set
1 + (f ^2) is V22() V23() ext-real Element of REAL
(1 + (f ^2)) ^2 is V22() V23() ext-real Element of REAL
K57((1 + (f ^2)),(1 + (f ^2))) is set
(2 * f) / ((1 + (f ^2)) ^2) is V22() V23() ext-real Element of REAL
Z is open V49() V50() V51() Element of K19(REAL)
x is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
(1 / 2) (#) x is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
dom ((1 / 2) (#) x) is set
((1 / 2) (#) x) `| Z is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
x is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
f is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
f + x is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
x / (f + x) is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
ln * (x / (f + x)) is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
dom x is set
dom (x / (f + x)) is set
x is set
dom (f + x) is set
dom x is set
(f + x) " {0} is set
(dom (f + x)) \ ((f + x) " {0}) is Element of K19((dom (f + x)))
K19((dom (f + x))) is set
(dom x) /\ ((dom (f + x)) \ ((f + x) " {0})) is Element of K19((dom (f + x)))
x + f is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
x / (x + f) is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
x is V22() V23() ext-real Element of REAL
(x / (x + f)) . x is V22() V23() ext-real Element of REAL
(x / (f + x)) . x is V22() V23() ext-real Element of REAL
x . x is V22() V23() ext-real Element of REAL
(f + x) . x is V22() V23() ext-real Element of REAL
((f + x) . x) " is V22() V23() ext-real Element of REAL
(x . x) * (((f + x) . x) ") is V22() V23() ext-real Element of REAL
(x . x) / ((f + x) . x) is V22() V23() ext-real Element of REAL
x ^2 is V22() V23() ext-real Element of REAL
K57(x,x) is set
1 + (x ^2) is V22() V23() ext-real Element of REAL
0 + 0 is V15() V16() V17() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() Element of NAT
x #Z 2 is V22() V23() ext-real Element of REAL
x |^ 2 is V22() V23() ext-real Element of REAL
f . x is V22() V23() ext-real Element of REAL
(f . x) + (x . x) is V22() V23() ext-real Element of REAL
x is V22() V23() ext-real Element of REAL
(x / (x + f)) . x is V22() V23() ext-real Element of REAL
x is V22() V23() ext-real Element of REAL
(((1 / 2) (#) x) `| Z) . x is V22() V23() ext-real Element of REAL
x ^2 is V22() V23() ext-real Element of REAL
K57(x,x) is set
1 + (x ^2) is V22() V23() ext-real Element of REAL
x * (1 + (x ^2)) is V22() V23() ext-real Element of REAL
1 / (x * (1 + (x ^2))) is V22() V23() ext-real Element of REAL
x . x is V22() V23() ext-real Element of REAL
x #Z 2 is V22() V23() ext-real Element of REAL
x |^ 2 is V22() V23() ext-real Element of REAL
(x / (x + f)) . x is V22() V23() ext-real Element of REAL
0 + 0 is V15() V16() V17() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() Element of NAT
(f + x) . x is V22() V23() ext-real Element of REAL
f . x is V22() V23() ext-real Element of REAL
(f . x) + (x . x) is V22() V23() ext-real Element of REAL
(x / (f + x)) . x is V22() V23() ext-real Element of REAL
((f + x) . x) " is V22() V23() ext-real Element of REAL
(x . x) * (((f + x) . x) ") is V22() V23() ext-real Element of REAL
(x ^2) / (1 + (x ^2)) is V22() V23() ext-real Element of REAL
diff ((ln * (x / (f + x))),x) is V22() V23() ext-real Element of REAL
(1 / 2) * (diff ((ln * (x / (f + x))),x)) is V22() V23() ext-real Element of REAL
diff ((x / (f + x)),x) is V22() V23() ext-real Element of REAL
(diff ((x / (f + x)),x)) / ((x / (f + x)) . x) is V22() V23() ext-real Element of REAL
(1 / 2) * ((diff ((x / (f + x)),x)) / ((x / (f + x)) . x)) is V22() V23() ext-real Element of REAL
(x / (f + x)) `| Z is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
((x / (f + x)) `| Z) . x is V22() V23() ext-real Element of REAL
(((x / (f + x)) `| Z) . x) / ((x / (f + x)) . x) is V22() V23() ext-real Element of REAL
(1 / 2) * ((((x / (f + x)) `| Z) . x) / ((x / (f + x)) . x)) is V22() V23() ext-real Element of REAL
2 * x is V22() V23() ext-real Element of REAL
(1 + (x ^2)) ^2 is V22() V23() ext-real Element of REAL
K57((1 + (x ^2)),(1 + (x ^2))) is set
(2 * x) / ((1 + (x ^2)) ^2) is V22() V23() ext-real Element of REAL
((2 * x) / ((1 + (x ^2)) ^2)) / ((x ^2) / (1 + (x ^2))) is V22() V23() ext-real Element of REAL
(1 / 2) * (((2 * x) / ((1 + (x ^2)) ^2)) / ((x ^2) / (1 + (x ^2)))) is V22() V23() ext-real Element of REAL
(1 / 2) * ((2 * x) / ((1 + (x ^2)) ^2)) is V22() V23() ext-real Element of REAL
((1 / 2) * ((2 * x) / ((1 + (x ^2)) ^2))) / ((x ^2) / (1 + (x ^2))) is V22() V23() ext-real Element of REAL
(1 / 2) * (2 * x) is V22() V23() ext-real Element of REAL
((1 / 2) * (2 * x)) / ((1 + (x ^2)) ^2) is V22() V23() ext-real Element of REAL
(((1 / 2) * (2 * x)) / ((1 + (x ^2)) ^2)) / ((x ^2) / (1 + (x ^2))) is V22() V23() ext-real Element of REAL
x / (1 + (x ^2)) is V22() V23() ext-real Element of REAL
(x / (1 + (x ^2))) / (1 + (x ^2)) is V22() V23() ext-real Element of REAL
((x / (1 + (x ^2))) / (1 + (x ^2))) / ((x ^2) / (1 + (x ^2))) is V22() V23() ext-real Element of REAL
(x / (1 + (x ^2))) / ((x ^2) / (1 + (x ^2))) is V22() V23() ext-real Element of REAL
((x / (1 + (x ^2))) / ((x ^2) / (1 + (x ^2)))) / (1 + (x ^2)) is V22() V23() ext-real Element of REAL
x / (x ^2) is V22() V23() ext-real Element of REAL
(x / (x ^2)) / (1 + (x ^2)) is V22() V23() ext-real Element of REAL
x / x is V22() V23() ext-real Element of REAL
(x / x) / x is V22() V23() ext-real Element of REAL
((x / x) / x) / (1 + (x ^2)) is V22() V23() ext-real Element of REAL
1 / x is V22() V23() ext-real Element of REAL
(1 / x) / (1 + (x ^2)) is V22() V23() ext-real Element of REAL
x is V22() V23() ext-real Element of REAL
(((1 / 2) (#) x) `| Z) . x is V22() V23() ext-real Element of REAL
x ^2 is V22() V23() ext-real Element of REAL
K57(x,x) is set
1 + (x ^2) is V22() V23() ext-real Element of REAL
x * (1 + (x ^2)) is V22() V23() ext-real Element of REAL
1 / (x * (1 + (x ^2))) is V22() V23() ext-real Element of REAL
Z is V15() V16() V17() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() Element of NAT
#Z Z is Relation-like REAL -defined REAL -valued V6() V30( REAL , REAL ) V39() V40() V41() Element of K19(K20(REAL,REAL))
ln * (#Z Z) is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
dom (ln * (#Z Z)) is set
x is open V49() V50() V51() Element of K19(REAL)
(ln * (#Z Z)) `| x is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
x is V22() V23() ext-real Element of REAL
(#Z Z) . x is V22() V23() ext-real Element of REAL
x #Z Z is V22() V23() ext-real Element of REAL
x is V22() V23() ext-real Element of REAL
((ln * (#Z Z)) `| x) . x is V22() V23() ext-real Element of REAL
Z / x is V22() V23() ext-real Element of REAL
diff ((#Z Z),x) is V22() V23() ext-real Element of REAL
Z - 1 is V22() V23() ext-real V68() Element of REAL
x #Z (Z - 1) is V22() V23() ext-real Element of REAL
Z * (x #Z (Z - 1)) is V22() V23() ext-real Element of REAL
x |^ Z is V22() V23() ext-real Element of REAL
(#Z Z) . x is V22() V23() ext-real Element of REAL
x #Z Z is V22() V23() ext-real Element of REAL
diff ((ln * (#Z Z)),x) is V22() V23() ext-real Element of REAL
(Z * (x #Z (Z - 1))) / (x #Z Z) is V22() V23() ext-real Element of REAL
(Z * (x #Z (Z - 1))) / (x |^ Z) is V22() V23() ext-real Element of REAL
x |^ 1 is V22() V23() ext-real Element of REAL
(x |^ Z) / (x |^ 1) is V22() V23() ext-real Element of REAL
Z * ((x |^ Z) / (x |^ 1)) is V22() V23() ext-real Element of REAL
(Z * ((x |^ Z) / (x |^ 1))) / (x |^ Z) is V22() V23() ext-real Element of REAL
((x |^ Z) / (x |^ 1)) / (x |^ Z) is V22() V23() ext-real Element of REAL
Z * (((x |^ Z) / (x |^ 1)) / (x |^ Z)) is V22() V23() ext-real Element of REAL
(x |^ Z) / (x |^ Z) is V22() V23() ext-real Element of REAL
((x |^ Z) / (x |^ Z)) / (x |^ 1) is V22() V23() ext-real Element of REAL
Z * (((x |^ Z) / (x |^ Z)) / (x |^ 1)) is V22() V23() ext-real Element of REAL
1 / (x |^ 1) is V22() V23() ext-real Element of REAL
Z * (1 / (x |^ 1)) is V22() V23() ext-real Element of REAL
1 / x is V22() V23() ext-real Element of REAL
Z * (1 / x) is V22() V23() ext-real Element of REAL
Z * 1 is V15() V16() V17() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() Element of NAT
(Z * 1) / x is V22() V23() ext-real Element of REAL
x is V22() V23() ext-real Element of REAL
((ln * (#Z Z)) `| x) . x is V22() V23() ext-real Element of REAL
Z / x is V22() V23() ext-real Element of REAL
Z is open V49() V50() V51() Element of K19(REAL)
x is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
x ^ is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
x is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
x / x is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
ln * (x / x) is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
(x ^) + (ln * (x / x)) is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
dom ((x ^) + (ln * (x / x))) is set
((x ^) + (ln * (x / x))) `| Z is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
dom (x ^) is set
dom (ln * (x / x)) is set
(dom (x ^)) /\ (dom (ln * (x / x))) is set
dom x is set
f is V22() V23() ext-real Element of REAL
x . f is V22() V23() ext-real Element of REAL
f - 1 is V22() V23() ext-real Element of REAL
x . f is V22() V23() ext-real Element of REAL
f - 0 is V22() V23() ext-real Element of REAL
f is V22() V23() ext-real Element of REAL
x . f is V22() V23() ext-real Element of REAL
0 + f is V22() V23() ext-real Element of REAL
f is V22() V23() ext-real Element of REAL
(((x ^) + (ln * (x / x))) `| Z) . f is V22() V23() ext-real Element of REAL
f ^2 is V22() V23() ext-real Element of REAL
K57(f,f) is set
f - 1 is V22() V23() ext-real Element of REAL
(f ^2) * (f - 1) is V22() V23() ext-real Element of REAL
1 / ((f ^2) * (f - 1)) is V22() V23() ext-real Element of REAL
x . f is V22() V23() ext-real Element of REAL
x . f is V22() V23() ext-real Element of REAL
diff ((x ^),f) is V22() V23() ext-real Element of REAL
diff ((ln * (x / x)),f) is V22() V23() ext-real Element of REAL
(diff ((x ^),f)) + (diff ((ln * (x / x)),f)) is V22() V23() ext-real Element of REAL
(x ^) `| Z is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
((x ^) `| Z) . f is V22() V23() ext-real Element of REAL
(((x ^) `| Z) . f) + (diff ((ln * (x / x)),f)) is V22() V23() ext-real Element of REAL
(ln * (x / x)) `| Z is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
((ln * (x / x)) `| Z) . f is V22() V23() ext-real Element of REAL
(((x ^) `| Z) . f) + (((ln * (x / x)) `| Z) . f) is V22() V23() ext-real Element of REAL
0 + f is V22() V23() ext-real Element of REAL
(0 + f) ^2 is V22() V23() ext-real Element of REAL
K57((0 + f),(0 + f)) is set
1 / ((0 + f) ^2) is V22() V23() ext-real Element of REAL
- (1 / ((0 + f) ^2)) is V22() V23() ext-real Element of REAL
(- (1 / ((0 + f) ^2))) + (((ln * (x / x)) `| Z) . f) is V22() V23() ext-real Element of REAL
1 - 0 is V22() V23() ext-real V68() Element of REAL
f - 0 is V22() V23() ext-real Element of REAL
(f - 1) * (f - 0) is V22() V23() ext-real Element of REAL
(1 - 0) / ((f - 1) * (f - 0)) is V22() V23() ext-real Element of REAL
(- (1 / ((0 + f) ^2))) + ((1 - 0) / ((f - 1) * (f - 0))) is V22() V23() ext-real Element of REAL
1 * (f - 1) is V22() V23() ext-real Element of REAL
(1 * (f - 1)) / ((f ^2) * (f - 1)) is V22() V23() ext-real Element of REAL
- ((1 * (f - 1)) / ((f ^2) * (f - 1))) is V22() V23() ext-real Element of REAL
(f - 1) * f is V22() V23() ext-real Element of REAL
1 / ((f - 1) * f) is V22() V23() ext-real Element of REAL
(- ((1 * (f - 1)) / ((f ^2) * (f - 1)))) + (1 / ((f - 1) * f)) is V22() V23() ext-real Element of REAL
1 * f is V22() V23() ext-real Element of REAL
((f - 1) * f) * f is V22() V23() ext-real Element of REAL
(1 * f) / (((f - 1) * f) * f) is V22() V23() ext-real Element of REAL
(- ((1 * (f - 1)) / ((f ^2) * (f - 1)))) + ((1 * f) / (((f - 1) * f) * f)) is V22() V23() ext-real Element of REAL
- (f - 1) is V22() V23() ext-real Element of REAL
(- (f - 1)) / ((f ^2) * (f - 1)) is V22() V23() ext-real Element of REAL
f / ((f ^2) * (f - 1)) is V22() V23() ext-real Element of REAL
((- (f - 1)) / ((f ^2) * (f - 1))) + (f / ((f ^2) * (f - 1))) is V22() V23() ext-real Element of REAL
- f is V22() V23() ext-real Element of REAL
(- f) + 1 is V22() V23() ext-real Element of REAL
((- f) + 1) + f is V22() V23() ext-real Element of REAL
(((- f) + 1) + f) / ((f ^2) * (f - 1)) is V22() V23() ext-real Element of REAL
f is V22() V23() ext-real Element of REAL
(((x ^) + (ln * (x / x))) `| Z) . f is V22() V23() ext-real Element of REAL
f ^2 is V22() V23() ext-real Element of REAL
K57(f,f) is set
f - 1 is V22() V23() ext-real Element of REAL
(f ^2) * (f - 1) is V22() V23() ext-real Element of REAL
1 / ((f ^2) * (f - 1)) is V22() V23() ext-real Element of REAL
Z is V22() V23() ext-real Element of REAL
log (number_e,Z) is V22() V23() ext-real Element of REAL
x is open V49() V50() V51() Element of K19(REAL)
x is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
exp_R * x is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
dom (exp_R * x) is set
(exp_R * x) `| x is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
dom x is set
f is set
f is V22() V23() ext-real Element of REAL
x . f is V22() V23() ext-real Element of REAL
(log (number_e,Z)) * f is V22() V23() ext-real Element of REAL
((log (number_e,Z)) * f) + 0 is V22() V23() ext-real Element of REAL
f is V22() V23() ext-real Element of REAL
f is V22() V23() ext-real Element of REAL
((exp_R * x) `| x) . f is V22() V23() ext-real Element of REAL
Z #R f is V22() V23() ext-real Element of REAL
(Z #R f) * (log (number_e,Z)) is V22() V23() ext-real Element of REAL
diff ((exp_R * x),f) is V22() V23() ext-real Element of REAL
x . f is V22() V23() ext-real Element of REAL
exp_R . (x . f) is V22() V23() ext-real Element of REAL
diff (x,f) is V22() V23() ext-real Element of REAL
(exp_R . (x . f)) * (diff (x,f)) is V22() V23() ext-real Element of REAL
x `| x is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
(x `| x) . f is V22() V23() ext-real Element of REAL
(exp_R . (x . f)) * ((x `| x) . f) is V22() V23() ext-real Element of REAL
(exp_R . (x . f)) * (log (number_e,Z)) is V22() V23() ext-real Element of REAL
f * (log (number_e,Z)) is V22() V23() ext-real Element of REAL
exp_R . (f * (log (number_e,Z))) is V22() V23() ext-real Element of REAL
(exp_R . (f * (log (number_e,Z)))) * (log (number_e,Z)) is V22() V23() ext-real Element of REAL
f is V22() V23() ext-real Element of REAL
((exp_R * x) `| x) . f is V22() V23() ext-real Element of REAL
Z #R f is V22() V23() ext-real Element of REAL
(Z #R f) * (log (number_e,Z)) is V22() V23() ext-real Element of REAL
Z is V22() V23() ext-real Element of REAL
log (number_e,Z) is V22() V23() ext-real Element of REAL
1 / (log (number_e,Z)) is V22() V23() ext-real Element of REAL
x is open V49() V50() V51() Element of K19(REAL)
x is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
exp_R * x is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
f is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
(exp_R * x) (#) f is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
(1 / (log (number_e,Z))) (#) ((exp_R * x) (#) f) is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
dom ((1 / (log (number_e,Z))) (#) ((exp_R * x) (#) f)) is set
((1 / (log (number_e,Z))) (#) ((exp_R * x) (#) f)) `| x is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
dom ((exp_R * x) (#) f) is set
dom (exp_R * x) is set
dom f is set
(dom (exp_R * x)) /\ (dom f) is set
- (1 / (log (number_e,Z))) is V22() V23() ext-real Element of REAL
x is V22() V23() ext-real Element of REAL
f . x is V22() V23() ext-real Element of REAL
1 * x is V22() V23() ext-real Element of REAL
(1 * x) + (- (1 / (log (number_e,Z)))) is V22() V23() ext-real Element of REAL
(1 * x) - (1 / (log (number_e,Z))) is V22() V23() ext-real Element of REAL
x is V22() V23() ext-real Element of REAL
x . x is V22() V23() ext-real Element of REAL
x * (log (number_e,Z)) is V22() V23() ext-real Element of REAL
log (number_e,1) is V22() V23() ext-real Element of REAL
number_e to_power (log (number_e,1)) is V22() V23() ext-real Element of REAL
x is V22() V23() ext-real Element of REAL
(((1 / (log (number_e,Z))) (#) ((exp_R * x) (#) f)) `| x) . x is V22() V23() ext-real Element of REAL
Z #R x is V22() V23() ext-real Element of REAL
x * (Z #R x) is V22() V23() ext-real Element of REAL
(exp_R * x) . x is V22() V23() ext-real Element of REAL
x . x is V22() V23() ext-real Element of REAL
exp_R . (x . x) is V22() V23() ext-real Element of REAL
x * (log (number_e,Z)) is V22() V23() ext-real Element of REAL
exp_R . (x * (log (number_e,Z))) is V22() V23() ext-real Element of REAL
diff (((exp_R * x) (#) f),x) is V22() V23() ext-real Element of REAL
(1 / (log (number_e,Z))) * (diff (((exp_R * x) (#) f),x)) is V22() V23() ext-real Element of REAL
((exp_R * x) (#) f) `| x is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
(((exp_R * x) (#) f) `| x) . x is V22() V23() ext-real Element of REAL
(1 / (log (number_e,Z))) * ((((exp_R * x) (#) f) `| x) . x) is V22() V23() ext-real Element of REAL
f . x is V22() V23() ext-real Element of REAL
diff ((exp_R * x),x) is V22() V23() ext-real Element of REAL
(f . x) * (diff ((exp_R * x),x)) is V22() V23() ext-real Element of REAL
diff (f,x) is V22() V23() ext-real Element of REAL
((exp_R * x) . x) * (diff (f,x)) is V22() V23() ext-real Element of REAL
((f . x) * (diff ((exp_R * x),x))) + (((exp_R * x) . x) * (diff (f,x))) is V22() V23() ext-real Element of REAL
(1 / (log (number_e,Z))) * (((f . x) * (diff ((exp_R * x),x))) + (((exp_R * x) . x) * (diff (f,x)))) is V22() V23() ext-real Element of REAL
(exp_R * x) `| x is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
((exp_R * x) `| x) . x is V22() V23() ext-real Element of REAL
(f . x) * (((exp_R * x) `| x) . x) is V22() V23() ext-real Element of REAL
((f . x) * (((exp_R * x) `| x) . x)) + (((exp_R * x) . x) * (diff (f,x))) is V22() V23() ext-real Element of REAL
(1 / (log (number_e,Z))) * (((f . x) * (((exp_R * x) `| x) . x)) + (((exp_R * x) . x) * (diff (f,x)))) is V22() V23() ext-real Element of REAL
f `| x is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
(f `| x) . x is V22() V23() ext-real Element of REAL
((exp_R * x) . x) * ((f `| x) . x) is V22() V23() ext-real Element of REAL
((f . x) * (((exp_R * x) `| x) . x)) + (((exp_R * x) . x) * ((f `| x) . x)) is V22() V23() ext-real Element of REAL
(1 / (log (number_e,Z))) * (((f . x) * (((exp_R * x) `| x) . x)) + (((exp_R * x) . x) * ((f `| x) . x))) is V22() V23() ext-real Element of REAL
(Z #R x) * (log (number_e,Z)) is V22() V23() ext-real Element of REAL
(f . x) * ((Z #R x) * (log (number_e,Z))) is V22() V23() ext-real Element of REAL
((f . x) * ((Z #R x) * (log (number_e,Z)))) + (((exp_R * x) . x) * ((f `| x) . x)) is V22() V23() ext-real Element of REAL
(1 / (log (number_e,Z))) * (((f . x) * ((Z #R x) * (log (number_e,Z)))) + (((exp_R * x) . x) * ((f `| x) . x))) is V22() V23() ext-real Element of REAL
((exp_R * x) . x) * 1 is V22() V23() ext-real Element of REAL
((f . x) * ((Z #R x) * (log (number_e,Z)))) + (((exp_R * x) . x) * 1) is V22() V23() ext-real Element of REAL
(1 / (log (number_e,Z))) * (((f . x) * ((Z #R x) * (log (number_e,Z)))) + (((exp_R * x) . x) * 1)) is V22() V23() ext-real Element of REAL
(f . x) * (log (number_e,Z)) is V22() V23() ext-real Element of REAL
((f . x) * (log (number_e,Z))) + 1 is V22() V23() ext-real Element of REAL
(((f . x) * (log (number_e,Z))) + 1) * (Z #R x) is V22() V23() ext-real Element of REAL
(1 / (log (number_e,Z))) * ((((f . x) * (log (number_e,Z))) + 1) * (Z #R x)) is V22() V23() ext-real Element of REAL
x - (1 / (log (number_e,Z))) is V22() V23() ext-real Element of REAL
(x - (1 / (log (number_e,Z)))) * (log (number_e,Z)) is V22() V23() ext-real Element of REAL
((x - (1 / (log (number_e,Z)))) * (log (number_e,Z))) + 1 is V22() V23() ext-real Element of REAL
(((x - (1 / (log (number_e,Z)))) * (log (number_e,Z))) + 1) * (Z #R x) is V22() V23() ext-real Element of REAL
(1 / (log (number_e,Z))) * ((((x - (1 / (log (number_e,Z)))) * (log (number_e,Z))) + 1) * (Z #R x)) is V22() V23() ext-real Element of REAL
(1 / (log (number_e,Z))) * (log (number_e,Z)) is V22() V23() ext-real Element of REAL
(x * (log (number_e,Z))) - ((1 / (log (number_e,Z))) * (log (number_e,Z))) is V22() V23() ext-real Element of REAL
((x * (log (number_e,Z))) - ((1 / (log (number_e,Z))) * (log (number_e,Z)))) + 1 is V22() V23() ext-real Element of REAL
(1 / (log (number_e,Z))) * (((x * (log (number_e,Z))) - ((1 / (log (number_e,Z))) * (log (number_e,Z)))) + 1) is V22() V23() ext-real Element of REAL
((1 / (log (number_e,Z))) * (((x * (log (number_e,Z))) - ((1 / (log (number_e,Z))) * (log (number_e,Z)))) + 1)) * (Z #R x) is V22() V23() ext-real Element of REAL
(x * (log (number_e,Z))) - 1 is V22() V23() ext-real Element of REAL
((x * (log (number_e,Z))) - 1) + 1 is V22() V23() ext-real Element of REAL
(1 / (log (number_e,Z))) * (((x * (log (number_e,Z))) - 1) + 1) is V22() V23() ext-real Element of REAL
((1 / (log (number_e,Z))) * (((x * (log (number_e,Z))) - 1) + 1)) * (Z #R x) is V22() V23() ext-real Element of REAL
((1 / (log (number_e,Z))) * (log (number_e,Z))) * x is V22() V23() ext-real Element of REAL
(((1 / (log (number_e,Z))) * (log (number_e,Z))) * x) * (Z #R x) is V22() V23() ext-real Element of REAL
1 * x is V22() V23() ext-real Element of REAL
(1 * x) * (Z #R x) is V22() V23() ext-real Element of REAL
x is V22() V23() ext-real Element of REAL
(((1 / (log (number_e,Z))) (#) ((exp_R * x) (#) f)) `| x) . x is V22() V23() ext-real Element of REAL
Z #R x is V22() V23() ext-real Element of REAL
x * (Z #R x) is V22() V23() ext-real Element of REAL
1 / number_e is V22() V23() ext-real Element of REAL
Z is V22() V23() ext-real Element of REAL
log (number_e,Z) is V22() V23() ext-real Element of REAL
1 + (log (number_e,Z)) is V22() V23() ext-real Element of REAL
1 / (1 + (log (number_e,Z))) is V22() V23() ext-real Element of REAL
x is open V49() V50() V51() Element of K19(REAL)
x is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
exp_R * x is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
(exp_R * x) (#) exp_R is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
(1 / (1 + (log (number_e,Z)))) (#) ((exp_R * x) (#) exp_R) is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
dom ((1 / (1 + (log (number_e,Z)))) (#) ((exp_R * x) (#) exp_R)) is set
((1 / (1 + (log (number_e,Z)))) (#) ((exp_R * x) (#) exp_R)) `| x is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
dom ((exp_R * x) (#) exp_R) is set
dom (exp_R * x) is set
dom exp_R is set
(dom (exp_R * x)) /\ (dom exp_R) is set
number_e * Z is V22() V23() ext-real Element of REAL
0 * Z is V22() V23() ext-real Element of REAL
log (number_e,1) is V22() V23() ext-real Element of REAL
log (number_e,number_e) is V22() V23() ext-real Element of REAL
(log (number_e,number_e)) + (log (number_e,Z)) is V22() V23() ext-real Element of REAL
log (number_e,(number_e * Z)) is V22() V23() ext-real Element of REAL
number_e to_power (log (number_e,1)) is V22() V23() ext-real Element of REAL
f is V22() V23() ext-real Element of REAL
(((1 / (1 + (log (number_e,Z)))) (#) ((exp_R * x) (#) exp_R)) `| x) . f is V22() V23() ext-real Element of REAL
Z #R f is V22() V23() ext-real Element of REAL
exp_R . f is V22() V23() ext-real Element of REAL
(Z #R f) * (exp_R . f) is V22() V23() ext-real Element of REAL
(exp_R * x) . f is V22() V23() ext-real Element of REAL
x . f is V22() V23() ext-real Element of REAL
exp_R . (x . f) is V22() V23() ext-real Element of REAL
f * (log (number_e,Z)) is V22() V23() ext-real Element of REAL
exp_R . (f * (log (number_e,Z))) is V22() V23() ext-real Element of REAL
diff (((exp_R * x) (#) exp_R),f) is V22() V23() ext-real Element of REAL
(1 / (1 + (log (number_e,Z)))) * (diff (((exp_R * x) (#) exp_R),f)) is V22() V23() ext-real Element of REAL
((exp_R * x) (#) exp_R) `| x is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
(((exp_R * x) (#) exp_R) `| x) . f is V22() V23() ext-real Element of REAL
(1 / (1 + (log (number_e,Z)))) * ((((exp_R * x) (#) exp_R) `| x) . f) is V22() V23() ext-real Element of REAL
diff ((exp_R * x),f) is V22() V23() ext-real Element of REAL
(exp_R . f) * (diff ((exp_R * x),f)) is V22() V23() ext-real Element of REAL
diff (exp_R,f) is V22() V23() ext-real Element of REAL
((exp_R * x) . f) * (diff (exp_R,f)) is V22() V23() ext-real Element of REAL
((exp_R . f) * (diff ((exp_R * x),f))) + (((exp_R * x) . f) * (diff (exp_R,f))) is V22() V23() ext-real Element of REAL
(1 / (1 + (log (number_e,Z)))) * (((exp_R . f) * (diff ((exp_R * x),f))) + (((exp_R * x) . f) * (diff (exp_R,f)))) is V22() V23() ext-real Element of REAL
(exp_R * x) `| x is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
((exp_R * x) `| x) . f is V22() V23() ext-real Element of REAL
(exp_R . f) * (((exp_R * x) `| x) . f) is V22() V23() ext-real Element of REAL
((exp_R . f) * (((exp_R * x) `| x) . f)) + (((exp_R * x) . f) * (diff (exp_R,f))) is V22() V23() ext-real Element of REAL
(1 / (1 + (log (number_e,Z)))) * (((exp_R . f) * (((exp_R * x) `| x) . f)) + (((exp_R * x) . f) * (diff (exp_R,f)))) is V22() V23() ext-real Element of REAL
((exp_R * x) . f) * (exp_R . f) is V22() V23() ext-real Element of REAL
((exp_R . f) * (((exp_R * x) `| x) . f)) + (((exp_R * x) . f) * (exp_R . f)) is V22() V23() ext-real Element of REAL
(1 / (1 + (log (number_e,Z)))) * (((exp_R . f) * (((exp_R * x) `| x) . f)) + (((exp_R * x) . f) * (exp_R . f))) is V22() V23() ext-real Element of REAL
(((exp_R * x) `| x) . f) + ((exp_R * x) . f) is V22() V23() ext-real Element of REAL
((((exp_R * x) `| x) . f) + ((exp_R * x) . f)) * (exp_R . f) is V22() V23() ext-real Element of REAL
(1 / (1 + (log (number_e,Z)))) * (((((exp_R * x) `| x) . f) + ((exp_R * x) . f)) * (exp_R . f)) is V22() V23() ext-real Element of REAL
(Z #R f) * (log (number_e,Z)) is V22() V23() ext-real Element of REAL
((Z #R f) * (log (number_e,Z))) + ((exp_R * x) . f) is V22() V23() ext-real Element of REAL
(((Z #R f) * (log (number_e,Z))) + ((exp_R * x) . f)) * (exp_R . f) is V22() V23() ext-real Element of REAL
(1 / (1 + (log (number_e,Z)))) * ((((Z #R f) * (log (number_e,Z))) + ((exp_R * x) . f)) * (exp_R . f)) is V22() V23() ext-real Element of REAL
(log (number_e,Z)) + 1 is V22() V23() ext-real Element of REAL
(1 / (1 + (log (number_e,Z)))) * ((log (number_e,Z)) + 1) is V22() V23() ext-real Element of REAL
((1 / (1 + (log (number_e,Z)))) * ((log (number_e,Z)) + 1)) * (Z #R f) is V22() V23() ext-real Element of REAL
(((1 / (1 + (log (number_e,Z)))) * ((log (number_e,Z)) + 1)) * (Z #R f)) * (exp_R . f) is V22() V23() ext-real Element of REAL
1 * (Z #R f) is V22() V23() ext-real Element of REAL
(1 * (Z #R f)) * (exp_R . f) is V22() V23() ext-real Element of REAL
f is V22() V23() ext-real Element of REAL
(((1 / (1 + (log (number_e,Z)))) (#) ((exp_R * x) (#) exp_R)) `| x) . f is V22() V23() ext-real Element of REAL
Z #R f is V22() V23() ext-real Element of REAL
exp_R . f is V22() V23() ext-real Element of REAL
(Z #R f) * (exp_R . f) is V22() V23() ext-real Element of REAL
Z is open V49() V50() V51() Element of K19(REAL)
x is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
exp_R * x is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
dom (exp_R * x) is set
(exp_R * x) `| Z is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
x is V22() V23() ext-real Element of REAL
x . x is V22() V23() ext-real Element of REAL
(- 1) * x is V22() V23() ext-real Element of REAL
((- 1) * x) + 0 is V22() V23() ext-real Element of REAL
- x is V22() V23() ext-real Element of REAL
dom x is set
x is set
x is V22() V23() ext-real Element of REAL
x is V22() V23() ext-real Element of REAL
((exp_R * x) `| Z) . x is V22() V23() ext-real Element of REAL
- x is V22() V23() ext-real Element of REAL
exp_R (- x) is V22() V23() ext-real Element of REAL
- (exp_R (- x)) is V22() V23() ext-real Element of REAL
diff ((exp_R * x),x) is V22() V23() ext-real Element of REAL
x . x is V22() V23() ext-real Element of REAL
exp_R . (x . x) is V22() V23() ext-real Element of REAL
diff (x,x) is V22() V23() ext-real Element of REAL
(exp_R . (x . x)) * (diff (x,x)) is V22() V23() ext-real Element of REAL
x `| Z is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
(x `| Z) . x is V22() V23() ext-real Element of REAL
(exp_R . (x . x)) * ((x `| Z) . x) is V22() V23() ext-real Element of REAL
(exp_R . (x . x)) * (- 1) is V22() V23() ext-real Element of REAL
exp_R . (- x) is V22() V23() ext-real Element of REAL
(exp_R . (- x)) * (- 1) is V22() V23() ext-real Element of REAL
- (exp_R . (- x)) is V22() V23() ext-real Element of REAL
x is V22() V23() ext-real Element of REAL
((exp_R * x) `| Z) . x is V22() V23() ext-real Element of REAL
- x is V22() V23() ext-real Element of REAL
exp_R (- x) is V22() V23() ext-real Element of REAL
- (exp_R (- x)) is V22() V23() ext-real Element of REAL
Z is open V49() V50() V51() Element of K19(REAL)
x is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
x is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
exp_R * x is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
x (#) (exp_R * x) is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
dom (x (#) (exp_R * x)) is set
(x (#) (exp_R * x)) `| Z is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
dom x is set
dom (exp_R * x) is set
(dom x) /\ (dom (exp_R * x)) is set
f is V22() V23() ext-real Element of REAL
x . f is V22() V23() ext-real Element of REAL
(- 1) * f is V22() V23() ext-real Element of REAL
((- 1) * f) + (- 1) is V22() V23() ext-real Element of REAL
- f is V22() V23() ext-real Element of REAL
(- f) - 1 is V22() V23() ext-real Element of REAL
f is V22() V23() ext-real Element of REAL
x . f is V22() V23() ext-real Element of REAL
- f is V22() V23() ext-real Element of REAL
f is V22() V23() ext-real Element of REAL
((x (#) (exp_R * x)) `| Z) . f is V22() V23() ext-real Element of REAL
exp_R f is V22() V23() ext-real Element of REAL
f / (exp_R f) is V22() V23() ext-real Element of REAL
(exp_R * x) . f is V22() V23() ext-real Element of REAL
x . f is V22() V23() ext-real Element of REAL
exp_R . (x . f) is V22() V23() ext-real Element of REAL
- f is V22() V23() ext-real Element of REAL
exp_R . (- f) is V22() V23() ext-real Element of REAL
exp_R (- f) is V22() V23() ext-real Element of REAL
diff (x,f) is V22() V23() ext-real Element of REAL
((exp_R * x) . f) * (diff (x,f)) is V22() V23() ext-real Element of REAL
x . f is V22() V23() ext-real Element of REAL
diff ((exp_R * x),f) is V22() V23() ext-real Element of REAL
(x . f) * (diff ((exp_R * x),f)) is V22() V23() ext-real Element of REAL
(((exp_R * x) . f) * (diff (x,f))) + ((x . f) * (diff ((exp_R * x),f))) is V22() V23() ext-real Element of REAL
(exp_R (- f)) * (diff (x,f)) is V22() V23() ext-real Element of REAL
(- f) - 1 is V22() V23() ext-real Element of REAL
((- f) - 1) * (diff ((exp_R * x),f)) is V22() V23() ext-real Element of REAL
((exp_R (- f)) * (diff (x,f))) + (((- f) - 1) * (diff ((exp_R * x),f))) is V22() V23() ext-real Element of REAL
x `| Z is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
(x `| Z) . f is V22() V23() ext-real Element of REAL
(exp_R (- f)) * ((x `| Z) . f) is V22() V23() ext-real Element of REAL
((exp_R (- f)) * ((x `| Z) . f)) + (((- f) - 1) * (diff ((exp_R * x),f))) is V22() V23() ext-real Element of REAL
(exp_R * x) `| Z is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
((exp_R * x) `| Z) . f is V22() V23() ext-real Element of REAL
((- f) - 1) * (((exp_R * x) `| Z) . f) is V22() V23() ext-real Element of REAL
((exp_R (- f)) * ((x `| Z) . f)) + (((- f) - 1) * (((exp_R * x) `| Z) . f)) is V22() V23() ext-real Element of REAL
(exp_R (- f)) * (- 1) is V22() V23() ext-real Element of REAL
((exp_R (- f)) * (- 1)) + (((- f) - 1) * (((exp_R * x) `| Z) . f)) is V22() V23() ext-real Element of REAL
- (exp_R (- f)) is V22() V23() ext-real Element of REAL
((- f) - 1) * (- (exp_R (- f))) is V22() V23() ext-real Element of REAL
((exp_R (- f)) * (- 1)) + (((- f) - 1) * (- (exp_R (- f)))) is V22() V23() ext-real Element of REAL
f * (exp_R (- f)) is V22() V23() ext-real Element of REAL
1 / (exp_R f) is V22() V23() ext-real Element of REAL
f * (1 / (exp_R f)) is V22() V23() ext-real Element of REAL
f is V22() V23() ext-real Element of REAL
((x (#) (exp_R * x)) `| Z) . f is V22() V23() ext-real Element of REAL
exp_R f is V22() V23() ext-real Element of REAL
f / (exp_R f) is V22() V23() ext-real Element of REAL
Z is V22() V23() ext-real Element of REAL
log (number_e,Z) is V22() V23() ext-real Element of REAL
x is open V49() V50() V51() Element of K19(REAL)
x is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
exp_R * x is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
- (exp_R * x) is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
K58(1) is V22() V23() V68() set
K58(1) (#) (exp_R * x) is Relation-like V6() set
dom (- (exp_R * x)) is set
(- (exp_R * x)) `| x is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
dom (exp_R * x) is set
dom x is set
f is set
- (log (number_e,Z)) is V22() V23() ext-real Element of REAL
f is V22() V23() ext-real Element of REAL
x . f is V22() V23() ext-real Element of REAL
(- (log (number_e,Z))) * f is V22() V23() ext-real Element of REAL
((- (log (number_e,Z))) * f) + 0 is V22() V23() ext-real Element of REAL
(log (number_e,Z)) * f is V22() V23() ext-real Element of REAL
- ((log (number_e,Z)) * f) is V22() V23() ext-real Element of REAL
f is V22() V23() ext-real Element of REAL
f is V22() V23() ext-real Element of REAL
((- (exp_R * x)) `| x) . f is V22() V23() ext-real Element of REAL
- f is V22() V23() ext-real Element of REAL
Z #R (- f) is V22() V23() ext-real Element of REAL
(Z #R (- f)) * (log (number_e,Z)) is V22() V23() ext-real Element of REAL
diff ((exp_R * x),f) is V22() V23() ext-real Element of REAL
(- 1) * (diff ((exp_R * x),f)) is V22() V23() ext-real Element of REAL
x . f is V22() V23() ext-real Element of REAL
exp_R . (x . f) is V22() V23() ext-real Element of REAL
diff (x,f) is V22() V23() ext-real Element of REAL
(exp_R . (x . f)) * (diff (x,f)) is V22() V23() ext-real Element of REAL
(- 1) * ((exp_R . (x . f)) * (diff (x,f))) is V22() V23() ext-real Element of REAL
x `| x is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
(x `| x) . f is V22() V23() ext-real Element of REAL
(exp_R . (x . f)) * ((x `| x) . f) is V22() V23() ext-real Element of REAL
(- 1) * ((exp_R . (x . f)) * ((x `| x) . f)) is V22() V23() ext-real Element of REAL
(exp_R . (x . f)) * (- (log (number_e,Z))) is V22() V23() ext-real Element of REAL
(- 1) * ((exp_R . (x . f)) * (- (log (number_e,Z)))) is V22() V23() ext-real Element of REAL
f * (log (number_e,Z)) is V22() V23() ext-real Element of REAL
- (f * (log (number_e,Z))) is V22() V23() ext-real Element of REAL
exp_R . (- (f * (log (number_e,Z)))) is V22() V23() ext-real Element of REAL
(exp_R . (- (f * (log (number_e,Z))))) * (- (log (number_e,Z))) is V22() V23() ext-real Element of REAL
(- 1) * ((exp_R . (- (f * (log (number_e,Z))))) * (- (log (number_e,Z)))) is V22() V23() ext-real Element of REAL
(Z #R (- f)) * (- (log (number_e,Z))) is V22() V23() ext-real Element of REAL
(- 1) * ((Z #R (- f)) * (- (log (number_e,Z)))) is V22() V23() ext-real Element of REAL
(- 1) (#) (exp_R * x) is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
dom ((- 1) (#) (exp_R * x)) is set
f is V22() V23() ext-real Element of REAL
((- (exp_R * x)) `| x) . f is V22() V23() ext-real Element of REAL
- f is V22() V23() ext-real Element of REAL
Z #R (- f) is V22() V23() ext-real Element of REAL
(Z #R (- f)) * (log (number_e,Z)) is V22() V23() ext-real Element of REAL
Z is V22() V23() ext-real Element of REAL
log (number_e,Z) is V22() V23() ext-real Element of REAL
1 / (log (number_e,Z)) is V22() V23() ext-real Element of REAL
x is open V49() V50() V51() Element of K19(REAL)
x is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
exp_R * x is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
- (exp_R * x) is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
K58(1) is V22() V23() V68() set
K58(1) (#) (exp_R * x) is Relation-like V6() set
f is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
(- (exp_R * x)) (#) f is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
(1 / (log (number_e,Z))) (#) ((- (exp_R * x)) (#) f) is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
dom ((1 / (log (number_e,Z))) (#) ((- (exp_R * x)) (#) f)) is set
((1 / (log (number_e,Z))) (#) ((- (exp_R * x)) (#) f)) `| x is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
x is V22() V23() ext-real Element of REAL
f . x is V22() V23() ext-real Element of REAL
1 * x is V22() V23() ext-real Element of REAL
(1 * x) + (1 / (log (number_e,Z))) is V22() V23() ext-real Element of REAL
dom ((- (exp_R * x)) (#) f) is set
dom (- (exp_R * x)) is set
dom f is set
(dom (- (exp_R * x))) /\ (dom f) is set
x is V22() V23() ext-real Element of REAL
x . x is V22() V23() ext-real Element of REAL
x * (log (number_e,Z)) is V22() V23() ext-real Element of REAL
- (x * (log (number_e,Z))) is V22() V23() ext-real Element of REAL
log (number_e,1) is V22() V23() ext-real Element of REAL
number_e to_power (log (number_e,1)) is V22() V23() ext-real Element of REAL
x is V22() V23() ext-real Element of REAL
(((1 / (log (number_e,Z))) (#) ((- (exp_R * x)) (#) f)) `| x) . x is V22() V23() ext-real Element of REAL
Z #R x is V22() V23() ext-real Element of REAL
x / (Z #R x) is V22() V23() ext-real Element of REAL
dom (exp_R * x) is set
(- (exp_R * x)) . x is V22() V23() ext-real Element of REAL
(exp_R * x) . x is V22() V23() ext-real Element of REAL
- ((exp_R * x) . x) is V22() V23() ext-real Element of REAL
x . x is V22() V23() ext-real Element of REAL
exp_R . (x . x) is V22() V23() ext-real Element of REAL
- (exp_R . (x . x)) is V22() V23() ext-real Element of REAL
x * (log (number_e,Z)) is V22() V23() ext-real Element of REAL
- (x * (log (number_e,Z))) is V22() V23() ext-real Element of REAL
exp_R . (- (x * (log (number_e,Z)))) is V22() V23() ext-real Element of REAL
- (exp_R . (- (x * (log (number_e,Z))))) is V22() V23() ext-real Element of REAL
- x is V22() V23() ext-real Element of REAL
Z #R (- x) is V22() V23() ext-real Element of REAL
- (Z #R (- x)) is V22() V23() ext-real Element of REAL
diff (((- (exp_R * x)) (#) f),x) is V22() V23() ext-real Element of REAL
(1 / (log (number_e,Z))) * (diff (((- (exp_R * x)) (#) f),x)) is V22() V23() ext-real Element of REAL
((- (exp_R * x)) (#) f) `| x is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
(((- (exp_R * x)) (#) f) `| x) . x is V22() V23() ext-real Element of REAL
(1 / (log (number_e,Z))) * ((((- (exp_R * x)) (#) f) `| x) . x) is V22() V23() ext-real Element of REAL
f . x is V22() V23() ext-real Element of REAL
diff ((- (exp_R * x)),x) is V22() V23() ext-real Element of REAL
(f . x) * (diff ((- (exp_R * x)),x)) is V22() V23() ext-real Element of REAL
diff (f,x) is V22() V23() ext-real Element of REAL
((- (exp_R * x)) . x) * (diff (f,x)) is V22() V23() ext-real Element of REAL
((f . x) * (diff ((- (exp_R * x)),x))) + (((- (exp_R * x)) . x) * (diff (f,x))) is V22() V23() ext-real Element of REAL
(1 / (log (number_e,Z))) * (((f . x) * (diff ((- (exp_R * x)),x))) + (((- (exp_R * x)) . x) * (diff (f,x)))) is V22() V23() ext-real Element of REAL
(- (exp_R * x)) `| x is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
((- (exp_R * x)) `| x) . x is V22() V23() ext-real Element of REAL
(f . x) * (((- (exp_R * x)) `| x) . x) is V22() V23() ext-real Element of REAL
((f . x) * (((- (exp_R * x)) `| x) . x)) + (((- (exp_R * x)) . x) * (diff (f,x))) is V22() V23() ext-real Element of REAL
(1 / (log (number_e,Z))) * (((f . x) * (((- (exp_R * x)) `| x) . x)) + (((- (exp_R * x)) . x) * (diff (f,x)))) is V22() V23() ext-real Element of REAL
f `| x is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
(f `| x) . x is V22() V23() ext-real Element of REAL
((- (exp_R * x)) . x) * ((f `| x) . x) is V22() V23() ext-real Element of REAL
((f . x) * (((- (exp_R * x)) `| x) . x)) + (((- (exp_R * x)) . x) * ((f `| x) . x)) is V22() V23() ext-real Element of REAL
(1 / (log (number_e,Z))) * (((f . x) * (((- (exp_R * x)) `| x) . x)) + (((- (exp_R * x)) . x) * ((f `| x) . x))) is V22() V23() ext-real Element of REAL
(Z #R (- x)) * (log (number_e,Z)) is V22() V23() ext-real Element of REAL
(f . x) * ((Z #R (- x)) * (log (number_e,Z))) is V22() V23() ext-real Element of REAL
((f . x) * ((Z #R (- x)) * (log (number_e,Z)))) + (((- (exp_R * x)) . x) * ((f `| x) . x)) is V22() V23() ext-real Element of REAL
(1 / (log (number_e,Z))) * (((f . x) * ((Z #R (- x)) * (log (number_e,Z)))) + (((- (exp_R * x)) . x) * ((f `| x) . x))) is V22() V23() ext-real Element of REAL
((- (exp_R * x)) . x) * 1 is V22() V23() ext-real Element of REAL
((f . x) * ((Z #R (- x)) * (log (number_e,Z)))) + (((- (exp_R * x)) . x) * 1) is V22() V23() ext-real Element of REAL
(1 / (log (number_e,Z))) * (((f . x) * ((Z #R (- x)) * (log (number_e,Z)))) + (((- (exp_R * x)) . x) * 1)) is V22() V23() ext-real Element of REAL
(f . x) * (log (number_e,Z)) is V22() V23() ext-real Element of REAL
((f . x) * (log (number_e,Z))) - 1 is V22() V23() ext-real Element of REAL
(((f . x) * (log (number_e,Z))) - 1) * (Z #R (- x)) is V22() V23() ext-real Element of REAL
(1 / (log (number_e,Z))) * ((((f . x) * (log (number_e,Z))) - 1) * (Z #R (- x))) is V22() V23() ext-real Element of REAL
x + (1 / (log (number_e,Z))) is V22() V23() ext-real Element of REAL
(x + (1 / (log (number_e,Z)))) * (log (number_e,Z)) is V22() V23() ext-real Element of REAL
((x + (1 / (log (number_e,Z)))) * (log (number_e,Z))) - 1 is V22() V23() ext-real Element of REAL
(((x + (1 / (log (number_e,Z)))) * (log (number_e,Z))) - 1) * (Z #R (- x)) is V22() V23() ext-real Element of REAL
(1 / (log (number_e,Z))) * ((((x + (1 / (log (number_e,Z)))) * (log (number_e,Z))) - 1) * (Z #R (- x))) is V22() V23() ext-real Element of REAL
(1 / (log (number_e,Z))) * (log (number_e,Z)) is V22() V23() ext-real Element of REAL
(x * (log (number_e,Z))) + ((1 / (log (number_e,Z))) * (log (number_e,Z))) is V22() V23() ext-real Element of REAL
((x * (log (number_e,Z))) + ((1 / (log (number_e,Z))) * (log (number_e,Z)))) - 1 is V22() V23() ext-real Element of REAL
(1 / (log (number_e,Z))) * (((x * (log (number_e,Z))) + ((1 / (log (number_e,Z))) * (log (number_e,Z)))) - 1) is V22() V23() ext-real Element of REAL
((1 / (log (number_e,Z))) * (((x * (log (number_e,Z))) + ((1 / (log (number_e,Z))) * (log (number_e,Z)))) - 1)) * (Z #R (- x)) is V22() V23() ext-real Element of REAL
(x * (log (number_e,Z))) + 1 is V22() V23() ext-real Element of REAL
((x * (log (number_e,Z))) + 1) - 1 is V22() V23() ext-real Element of REAL
(1 / (log (number_e,Z))) * (((x * (log (number_e,Z))) + 1) - 1) is V22() V23() ext-real Element of REAL
((1 / (log (number_e,Z))) * (((x * (log (number_e,Z))) + 1) - 1)) * (Z #R (- x)) is V22() V23() ext-real Element of REAL
((1 / (log (number_e,Z))) * (log (number_e,Z))) * x is V22() V23() ext-real Element of REAL
(((1 / (log (number_e,Z))) * (log (number_e,Z))) * x) * (Z #R (- x)) is V22() V23() ext-real Element of REAL
1 * x is V22() V23() ext-real Element of REAL
(1 * x) * (Z #R (- x)) is V22() V23() ext-real Element of REAL
1 / (Z #R x) is V22() V23() ext-real Element of REAL
x * (1 / (Z #R x)) is V22() V23() ext-real Element of REAL
x is V22() V23() ext-real Element of REAL
(((1 / (log (number_e,Z))) (#) ((- (exp_R * x)) (#) f)) `| x) . x is V22() V23() ext-real Element of REAL
Z #R x is V22() V23() ext-real Element of REAL
x / (Z #R x) is V22() V23() ext-real Element of REAL
Z is V22() V23() ext-real Element of REAL
log (number_e,Z) is V22() V23() ext-real Element of REAL
(log (number_e,Z)) - 1 is V22() V23() ext-real Element of REAL
1 / ((log (number_e,Z)) - 1) is V22() V23() ext-real Element of REAL
x is open V49() V50() V51() Element of K19(REAL)
x is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
exp_R * x is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
(exp_R * x) / exp_R is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
(1 / ((log (number_e,Z)) - 1)) (#) ((exp_R * x) / exp_R) is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
dom ((1 / ((log (number_e,Z)) - 1)) (#) ((exp_R * x) / exp_R)) is set
((1 / ((log (number_e,Z)) - 1)) (#) ((exp_R * x) / exp_R)) `| x is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
dom ((exp_R * x) / exp_R) is set
dom exp_R is set
dom (exp_R * x) is set
exp_R " {0} is set
(dom exp_R) \ (exp_R " {0}) is Element of K19((dom exp_R))
K19((dom exp_R)) is set
(dom (exp_R * x)) /\ ((dom exp_R) \ (exp_R " {0})) is Element of K19((dom exp_R))
f is V22() V23() ext-real Element of REAL
exp_R . f is V22() V23() ext-real Element of REAL
log (number_e,number_e) is V22() V23() ext-real Element of REAL
number_e to_power (log (number_e,number_e)) is V22() V23() ext-real Element of REAL
f is V22() V23() ext-real Element of REAL
(((1 / ((log (number_e,Z)) - 1)) (#) ((exp_R * x) / exp_R)) `| x) . f is V22() V23() ext-real Element of REAL
Z #R f is V22() V23() ext-real Element of REAL
exp_R . f is V22() V23() ext-real Element of REAL
(Z #R f) / (exp_R . f) is V22() V23() ext-real Element of REAL
(exp_R * x) . f is V22() V23() ext-real Element of REAL
x . f is V22() V23() ext-real Element of REAL
exp_R . (x . f) is V22() V23() ext-real Element of REAL
f * (log (number_e,Z)) is V22() V23() ext-real Element of REAL
exp_R . (f * (log (number_e,Z))) is V22() V23() ext-real Element of REAL
diff (((exp_R * x) / exp_R),f) is V22() V23() ext-real Element of REAL
(1 / ((log (number_e,Z)) - 1)) * (diff (((exp_R * x) / exp_R),f)) is V22() V23() ext-real Element of REAL
diff ((exp_R * x),f) is V22() V23() ext-real Element of REAL
(diff ((exp_R * x),f)) * (exp_R . f) is V22() V23() ext-real Element of REAL
diff (exp_R,f) is V22() V23() ext-real Element of REAL
(diff (exp_R,f)) * ((exp_R * x) . f) is V22() V23() ext-real Element of REAL
((diff ((exp_R * x),f)) * (exp_R . f)) - ((diff (exp_R,f)) * ((exp_R * x) . f)) is V22() V23() ext-real Element of REAL
(exp_R . f) ^2 is V22() V23() ext-real Element of REAL
K57((exp_R . f),(exp_R . f)) is set
(((diff ((exp_R * x),f)) * (exp_R . f)) - ((diff (exp_R,f)) * ((exp_R * x) . f))) / ((exp_R . f) ^2) is V22() V23() ext-real Element of REAL
(1 / ((log (number_e,Z)) - 1)) * ((((diff ((exp_R * x),f)) * (exp_R . f)) - ((diff (exp_R,f)) * ((exp_R * x) . f))) / ((exp_R . f) ^2)) is V22() V23() ext-real Element of REAL
(exp_R . f) * (Z #R f) is V22() V23() ext-real Element of REAL
((diff ((exp_R * x),f)) * (exp_R . f)) - ((exp_R . f) * (Z #R f)) is V22() V23() ext-real Element of REAL
(((diff ((exp_R * x),f)) * (exp_R . f)) - ((exp_R . f) * (Z #R f))) / ((exp_R . f) ^2) is V22() V23() ext-real Element of REAL
(1 / ((log (number_e,Z)) - 1)) * ((((diff ((exp_R * x),f)) * (exp_R . f)) - ((exp_R . f) * (Z #R f))) / ((exp_R . f) ^2)) is V22() V23() ext-real Element of REAL
(diff ((exp_R * x),f)) - (Z #R f) is V22() V23() ext-real Element of REAL
((diff ((exp_R * x),f)) - (Z #R f)) * (exp_R . f) is V22() V23() ext-real Element of REAL
(exp_R . f) * (exp_R . f) is V22() V23() ext-real Element of REAL
(((diff ((exp_R * x),f)) - (Z #R f)) * (exp_R . f)) / ((exp_R . f) * (exp_R . f)) is V22() V23() ext-real Element of REAL
(1 / ((log (number_e,Z)) - 1)) * ((((diff ((exp_R * x),f)) - (Z #R f)) * (exp_R . f)) / ((exp_R . f) * (exp_R . f))) is V22() V23() ext-real Element of REAL
((diff ((exp_R * x),f)) - (Z #R f)) / (exp_R . f) is V22() V23() ext-real Element of REAL
(1 / ((log (number_e,Z)) - 1)) * (((diff ((exp_R * x),f)) - (Z #R f)) / (exp_R . f)) is V22() V23() ext-real Element of REAL
(1 / ((log (number_e,Z)) - 1)) * ((diff ((exp_R * x),f)) - (Z #R f)) is V22() V23() ext-real Element of REAL
((1 / ((log (number_e,Z)) - 1)) * ((diff ((exp_R * x),f)) - (Z #R f))) / (exp_R . f) is V22() V23() ext-real Element of REAL
(exp_R * x) `| x is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
((exp_R * x) `| x) . f is V22() V23() ext-real Element of REAL
(((exp_R * x) `| x) . f) - (Z #R f) is V22() V23() ext-real Element of REAL
(1 / ((log (number_e,Z)) - 1)) * ((((exp_R * x) `| x) . f) - (Z #R f)) is V22() V23() ext-real Element of REAL
((1 / ((log (number_e,Z)) - 1)) * ((((exp_R * x) `| x) . f) - (Z #R f))) / (exp_R . f) is V22() V23() ext-real Element of REAL
(Z #R f) * (log (number_e,Z)) is V22() V23() ext-real Element of REAL
((Z #R f) * (log (number_e,Z))) - (Z #R f) is V22() V23() ext-real Element of REAL
(1 / ((log (number_e,Z)) - 1)) * (((Z #R f) * (log (number_e,Z))) - (Z #R f)) is V22() V23() ext-real Element of REAL
((1 / ((log (number_e,Z)) - 1)) * (((Z #R f) * (log (number_e,Z))) - (Z #R f))) / (exp_R . f) is V22() V23() ext-real Element of REAL
(1 / ((log (number_e,Z)) - 1)) * ((log (number_e,Z)) - 1) is V22() V23() ext-real Element of REAL
((1 / ((log (number_e,Z)) - 1)) * ((log (number_e,Z)) - 1)) * (Z #R f) is V22() V23() ext-real Element of REAL
(((1 / ((log (number_e,Z)) - 1)) * ((log (number_e,Z)) - 1)) * (Z #R f)) / (exp_R . f) is V22() V23() ext-real Element of REAL
1 * (Z #R f) is V22() V23() ext-real Element of REAL
(1 * (Z #R f)) / (exp_R . f) is V22() V23() ext-real Element of REAL
f is V22() V23() ext-real Element of REAL
(((1 / ((log (number_e,Z)) - 1)) (#) ((exp_R * x) / exp_R)) `| x) . f is V22() V23() ext-real Element of REAL
Z #R f is V22() V23() ext-real Element of REAL
exp_R . f is V22() V23() ext-real Element of REAL
(Z #R f) / (exp_R . f) is V22() V23() ext-real Element of REAL
Z is V22() V23() ext-real Element of REAL
log (number_e,Z) is V22() V23() ext-real Element of REAL
1 - (log (number_e,Z)) is V22() V23() ext-real Element of REAL
1 / (1 - (log (number_e,Z))) is V22() V23() ext-real Element of REAL
x is open V49() V50() V51() Element of K19(REAL)
x is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
exp_R * x is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
exp_R / (exp_R * x) is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
(1 / (1 - (log (number_e,Z)))) (#) (exp_R / (exp_R * x)) is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
dom ((1 / (1 - (log (number_e,Z)))) (#) (exp_R / (exp_R * x))) is set
((1 / (1 - (log (number_e,Z)))) (#) (exp_R / (exp_R * x))) `| x is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
dom (exp_R / (exp_R * x)) is set
dom (exp_R * x) is set
dom exp_R is set
(exp_R * x) " {0} is set
(dom (exp_R * x)) \ ((exp_R * x) " {0}) is Element of K19((dom (exp_R * x)))
K19((dom (exp_R * x))) is set
(dom exp_R) /\ ((dom (exp_R * x)) \ ((exp_R * x) " {0})) is Element of K19((dom (exp_R * x)))
f is V22() V23() ext-real Element of REAL
(exp_R * x) . f is V22() V23() ext-real Element of REAL
x . f is V22() V23() ext-real Element of REAL
exp_R . (x . f) is V22() V23() ext-real Element of REAL
log (number_e,number_e) is V22() V23() ext-real Element of REAL
number_e to_power (log (number_e,number_e)) is V22() V23() ext-real Element of REAL
f is V22() V23() ext-real Element of REAL
(((1 / (1 - (log (number_e,Z)))) (#) (exp_R / (exp_R * x))) `| x) . f is V22() V23() ext-real Element of REAL
exp_R . f is V22() V23() ext-real Element of REAL
Z #R f is V22() V23() ext-real Element of REAL
(exp_R . f) / (Z #R f) is V22() V23() ext-real Element of REAL
(exp_R * x) . f is V22() V23() ext-real Element of REAL
x . f is V22() V23() ext-real Element of REAL
exp_R . (x . f) is V22() V23() ext-real Element of REAL
f * (log (number_e,Z)) is V22() V23() ext-real Element of REAL
exp_R . (f * (log (number_e,Z))) is V22() V23() ext-real Element of REAL
diff ((exp_R / (exp_R * x)),f) is V22() V23() ext-real Element of REAL
(1 / (1 - (log (number_e,Z)))) * (diff ((exp_R / (exp_R * x)),f)) is V22() V23() ext-real Element of REAL
diff (exp_R,f) is V22() V23() ext-real Element of REAL
(diff (exp_R,f)) * ((exp_R * x) . f) is V22() V23() ext-real Element of REAL
diff ((exp_R * x),f) is V22() V23() ext-real Element of REAL
(diff ((exp_R * x),f)) * (exp_R . f) is V22() V23() ext-real Element of REAL
((diff (exp_R,f)) * ((exp_R * x) . f)) - ((diff ((exp_R * x),f)) * (exp_R . f)) is V22() V23() ext-real Element of REAL
((exp_R * x) . f) ^2 is V22() V23() ext-real Element of REAL
K57(((exp_R * x) . f),((exp_R * x) . f)) is set
(((diff (exp_R,f)) * ((exp_R * x) . f)) - ((diff ((exp_R * x),f)) * (exp_R . f))) / (((exp_R * x) . f) ^2) is V22() V23() ext-real Element of REAL
(1 / (1 - (log (number_e,Z)))) * ((((diff (exp_R,f)) * ((exp_R * x) . f)) - ((diff ((exp_R * x),f)) * (exp_R . f))) / (((exp_R * x) . f) ^2)) is V22() V23() ext-real Element of REAL
(exp_R . f) * (Z #R f) is V22() V23() ext-real Element of REAL
((exp_R . f) * (Z #R f)) - ((diff ((exp_R * x),f)) * (exp_R . f)) is V22() V23() ext-real Element of REAL
(Z #R f) ^2 is V22() V23() ext-real Element of REAL
K57((Z #R f),(Z #R f)) is set
(((exp_R . f) * (Z #R f)) - ((diff ((exp_R * x),f)) * (exp_R . f))) / ((Z #R f) ^2) is V22() V23() ext-real Element of REAL
(1 / (1 - (log (number_e,Z)))) * ((((exp_R . f) * (Z #R f)) - ((diff ((exp_R * x),f)) * (exp_R . f))) / ((Z #R f) ^2)) is V22() V23() ext-real Element of REAL
(Z #R f) - (diff ((exp_R * x),f)) is V22() V23() ext-real Element of REAL
(exp_R . f) * ((Z #R f) - (diff ((exp_R * x),f))) is V22() V23() ext-real Element of REAL
((exp_R . f) * ((Z #R f) - (diff ((exp_R * x),f)))) / ((Z #R f) ^2) is V22() V23() ext-real Element of REAL
(1 / (1 - (log (number_e,Z)))) * (((exp_R . f) * ((Z #R f) - (diff ((exp_R * x),f)))) / ((Z #R f) ^2)) is V22() V23() ext-real Element of REAL
(exp_R * x) `| x is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
((exp_R * x) `| x) . f is V22() V23() ext-real Element of REAL
(Z #R f) - (((exp_R * x) `| x) . f) is V22() V23() ext-real Element of REAL
(exp_R . f) * ((Z #R f) - (((exp_R * x) `| x) . f)) is V22() V23() ext-real Element of REAL
((exp_R . f) * ((Z #R f) - (((exp_R * x) `| x) . f))) / ((Z #R f) ^2) is V22() V23() ext-real Element of REAL
(1 / (1 - (log (number_e,Z)))) * (((exp_R . f) * ((Z #R f) - (((exp_R * x) `| x) . f))) / ((Z #R f) ^2)) is V22() V23() ext-real Element of REAL
(Z #R f) * (log (number_e,Z)) is V22() V23() ext-real Element of REAL
(Z #R f) - ((Z #R f) * (log (number_e,Z))) is V22() V23() ext-real Element of REAL
(exp_R . f) * ((Z #R f) - ((Z #R f) * (log (number_e,Z)))) is V22() V23() ext-real Element of REAL
((exp_R . f) * ((Z #R f) - ((Z #R f) * (log (number_e,Z))))) / ((Z #R f) ^2) is V22() V23() ext-real Element of REAL
(1 / (1 - (log (number_e,Z)))) * (((exp_R . f) * ((Z #R f) - ((Z #R f) * (log (number_e,Z))))) / ((Z #R f) ^2)) is V22() V23() ext-real Element of REAL
(1 - (log (number_e,Z))) * (exp_R . f) is V22() V23() ext-real Element of REAL
((1 - (log (number_e,Z))) * (exp_R . f)) * (Z #R f) is V22() V23() ext-real Element of REAL
(1 / (1 - (log (number_e,Z)))) * (((1 - (log (number_e,Z))) * (exp_R . f)) * (Z #R f)) is V22() V23() ext-real Element of REAL
((1 / (1 - (log (number_e,Z)))) * (((1 - (log (number_e,Z))) * (exp_R . f)) * (Z #R f))) / ((Z #R f) ^2) is V22() V23() ext-real Element of REAL
(1 / (1 - (log (number_e,Z)))) * (1 - (log (number_e,Z))) is V22() V23() ext-real Element of REAL
((1 / (1 - (log (number_e,Z)))) * (1 - (log (number_e,Z)))) * (exp_R . f) is V22() V23() ext-real Element of REAL
(((1 / (1 - (log (number_e,Z)))) * (1 - (log (number_e,Z)))) * (exp_R . f)) * (Z #R f) is V22() V23() ext-real Element of REAL
((((1 / (1 - (log (number_e,Z)))) * (1 - (log (number_e,Z)))) * (exp_R . f)) * (Z #R f)) / ((Z #R f) ^2) is V22() V23() ext-real Element of REAL
1 * (exp_R . f) is V22() V23() ext-real Element of REAL
(1 * (exp_R . f)) * (Z #R f) is V22() V23() ext-real Element of REAL
((1 * (exp_R . f)) * (Z #R f)) / ((Z #R f) ^2) is V22() V23() ext-real Element of REAL
f is V22() V23() ext-real Element of REAL
(((1 / (1 - (log (number_e,Z)))) (#) (exp_R / (exp_R * x))) `| x) . f is V22() V23() ext-real Element of REAL
exp_R . f is V22() V23() ext-real Element of REAL
Z #R f is V22() V23() ext-real Element of REAL
(exp_R . f) / (Z #R f) is V22() V23() ext-real Element of REAL
Z is open V49() V50() V51() Element of K19(REAL)
x is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
exp_R + x is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
ln * (exp_R + x) is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
dom (ln * (exp_R + x)) is set
(ln * (exp_R + x)) `| Z is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
x is V22() V23() ext-real Element of REAL
x . x is V22() V23() ext-real Element of REAL
0 * x is V22() V23() ext-real Element of REAL
(0 * x) + 1 is V22() V23() ext-real Element of REAL
dom (exp_R + x) is set
x is set
dom exp_R is set
dom x is set
(dom exp_R) /\ (dom x) is set
(exp_R + x) `| Z is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
x is V22() V23() ext-real Element of REAL
((exp_R + x) `| Z) . x is V22() V23() ext-real Element of REAL
exp_R . x is V22() V23() ext-real Element of REAL
diff (exp_R,x) is V22() V23() ext-real Element of REAL
diff (x,x) is V22() V23() ext-real Element of REAL
(diff (exp_R,x)) + (diff (x,x)) is V22() V23() ext-real Element of REAL
(exp_R . x) + (diff (x,x)) is V22() V23() ext-real Element of REAL
x `| Z is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
(x `| Z) . x is V22() V23() ext-real Element of REAL
(exp_R . x) + ((x `| Z) . x) is V22() V23() ext-real Element of REAL
(exp_R . x) + 0 is V22() V23() ext-real Element of REAL
x is V22() V23() ext-real Element of REAL
(exp_R + x) . x is V22() V23() ext-real Element of REAL
exp_R . x is V22() V23() ext-real Element of REAL
x . x is V22() V23() ext-real Element of REAL
(exp_R . x) + (x . x) is V22() V23() ext-real Element of REAL
(exp_R . x) + 1 is V22() V23() ext-real Element of REAL
x is V22() V23() ext-real Element of REAL
(exp_R + x) . x is V22() V23() ext-real Element of REAL
x is V22() V23() ext-real Element of REAL
((ln * (exp_R + x)) `| Z) . x is V22() V23() ext-real Element of REAL
exp_R . x is V22() V23() ext-real Element of REAL
(exp_R . x) + 1 is V22() V23() ext-real Element of REAL
(exp_R . x) / ((exp_R . x) + 1) is V22() V23() ext-real Element of REAL
(exp_R + x) . x is V22() V23() ext-real Element of REAL
x . x is V22() V23() ext-real Element of REAL
(exp_R . x) + (x . x) is V22() V23() ext-real Element of REAL
diff ((ln * (exp_R + x)),x) is V22() V23() ext-real Element of REAL
diff ((exp_R + x),x) is V22() V23() ext-real Element of REAL
(diff ((exp_R + x),x)) / ((exp_R + x) . x) is V22() V23() ext-real Element of REAL
((exp_R + x) `| Z) . x is V22() V23() ext-real Element of REAL
(((exp_R + x) `| Z) . x) / ((exp_R + x) . x) is V22() V23() ext-real Element of REAL
x is V22() V23() ext-real Element of REAL
((ln * (exp_R + x)) `| Z) . x is V22() V23() ext-real Element of REAL
exp_R . x is V22() V23() ext-real Element of REAL
(exp_R . x) + 1 is V22() V23() ext-real Element of REAL
(exp_R . x) / ((exp_R . x) + 1) is V22() V23() ext-real Element of REAL
Z is open V49() V50() V51() Element of K19(REAL)
x is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
exp_R - x is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
- x is Relation-like V6() V39() set
K58(1) is V22() V23() V68() set
K58(1) (#) x is Relation-like V6() set
exp_R + (- x) is Relation-like V6() set
ln * (exp_R - x) is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
dom (ln * (exp_R - x)) is set
(ln * (exp_R - x)) `| Z is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
x is V22() V23() ext-real Element of REAL
x . x is V22() V23() ext-real Element of REAL
0 * x is V22() V23() ext-real Element of REAL
(0 * x) + 1 is V22() V23() ext-real Element of REAL
dom (exp_R - x) is set
x is set
dom exp_R is set
dom x is set
(dom exp_R) /\ (dom x) is set
(exp_R - x) `| Z is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
x is V22() V23() ext-real Element of REAL
((exp_R - x) `| Z) . x is V22() V23() ext-real Element of REAL
exp_R . x is V22() V23() ext-real Element of REAL
diff (exp_R,x) is V22() V23() ext-real Element of REAL
diff (x,x) is V22() V23() ext-real Element of REAL
(diff (exp_R,x)) - (diff (x,x)) is V22() V23() ext-real Element of REAL
(exp_R . x) - (diff (x,x)) is V22() V23() ext-real Element of REAL
x `| Z is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
(x `| Z) . x is V22() V23() ext-real Element of REAL
(exp_R . x) - ((x `| Z) . x) is V22() V23() ext-real Element of REAL
(exp_R . x) - 0 is V22() V23() ext-real Element of REAL
x is V22() V23() ext-real Element of REAL
(exp_R - x) . x is V22() V23() ext-real Element of REAL
x is V22() V23() ext-real Element of REAL
((ln * (exp_R - x)) `| Z) . x is V22() V23() ext-real Element of REAL
exp_R . x is V22() V23() ext-real Element of REAL
(exp_R . x) - 1 is V22() V23() ext-real Element of REAL
(exp_R . x) / ((exp_R . x) - 1) is V22() V23() ext-real Element of REAL
(exp_R - x) . x is V22() V23() ext-real Element of REAL
x . x is V22() V23() ext-real Element of REAL
(exp_R . x) - (x . x) is V22() V23() ext-real Element of REAL
diff ((ln * (exp_R - x)),x) is V22() V23() ext-real Element of REAL
diff ((exp_R - x),x) is V22() V23() ext-real Element of REAL
(diff ((exp_R - x),x)) / ((exp_R - x) . x) is V22() V23() ext-real Element of REAL
((exp_R - x) `| Z) . x is V22() V23() ext-real Element of REAL
(((exp_R - x) `| Z) . x) / ((exp_R - x) . x) is V22() V23() ext-real Element of REAL
x is V22() V23() ext-real Element of REAL
((ln * (exp_R - x)) `| Z) . x is V22() V23() ext-real Element of REAL
exp_R . x is V22() V23() ext-real Element of REAL
(exp_R . x) - 1 is V22() V23() ext-real Element of REAL
(exp_R . x) / ((exp_R . x) - 1) is V22() V23() ext-real Element of REAL
Z is open V49() V50() V51() Element of K19(REAL)
x is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
x - exp_R is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
- exp_R is Relation-like V6() V39() set
K58(1) is V22() V23() V68() set
K58(1) (#) exp_R is Relation-like V6() set
x + (- exp_R) is Relation-like V6() set
ln * (x - exp_R) is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
- (ln * (x - exp_R)) is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
K58(1) (#) (ln * (x - exp_R)) is Relation-like V6() set
dom (- (ln * (x - exp_R))) is set
(- (ln * (x - exp_R))) `| Z is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
x is V22() V23() ext-real Element of REAL
x . x is V22() V23() ext-real Element of REAL
0 * x is V22() V23() ext-real Element of REAL
(0 * x) + 1 is V22() V23() ext-real Element of REAL
dom (ln * (x - exp_R)) is set
dom (x - exp_R) is set
x is set
dom exp_R is set
dom x is set
(dom exp_R) /\ (dom x) is set
x is V22() V23() ext-real Element of REAL
(x - exp_R) . x is V22() V23() ext-real Element of REAL
(x - exp_R) `| Z is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
x is V22() V23() ext-real Element of REAL
((x - exp_R) `| Z) . x is V22() V23() ext-real Element of REAL
exp_R . x is V22() V23() ext-real Element of REAL
- (exp_R . x) is V22() V23() ext-real Element of REAL
diff (x,x) is V22() V23() ext-real Element of REAL
diff (exp_R,x) is V22() V23() ext-real Element of REAL
(diff (x,x)) - (diff (exp_R,x)) is V22() V23() ext-real Element of REAL
(diff (x,x)) - (exp_R . x) is V22() V23() ext-real Element of REAL
x `| Z is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
(x `| Z) . x is V22() V23() ext-real Element of REAL
((x `| Z) . x) - (exp_R . x) is V22() V23() ext-real Element of REAL
0 - (exp_R . x) is V22() V23() ext-real Element of REAL
x is V22() V23() ext-real Element of REAL
((- (ln * (x - exp_R))) `| Z) . x is V22() V23() ext-real Element of REAL
exp_R . x is V22() V23() ext-real Element of REAL
1 - (exp_R . x) is V22() V23() ext-real Element of REAL
(exp_R . x) / (1 - (exp_R . x)) is V22() V23() ext-real Element of REAL
(x - exp_R) . x is V22() V23() ext-real Element of REAL
x . x is V22() V23() ext-real Element of REAL
(x . x) - (exp_R . x) is V22() V23() ext-real Element of REAL
(- 1) (#) (ln * (x - exp_R)) is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
((- 1) (#) (ln * (x - exp_R))) `| Z is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
(((- 1) (#) (ln * (x - exp_R))) `| Z) . x is V22() V23() ext-real Element of REAL
diff ((ln * (x - exp_R)),x) is V22() V23() ext-real Element of REAL
(- 1) * (diff ((ln * (x - exp_R)),x)) is V22() V23() ext-real Element of REAL
diff ((x - exp_R),x) is V22() V23() ext-real Element of REAL
(diff ((x - exp_R),x)) / ((x - exp_R) . x) is V22() V23() ext-real Element of REAL
(- 1) * ((diff ((x - exp_R),x)) / ((x - exp_R) . x)) is V22() V23() ext-real Element of REAL
((x - exp_R) `| Z) . x is V22() V23() ext-real Element of REAL
(((x - exp_R) `| Z) . x) / ((x - exp_R) . x) is V22() V23() ext-real Element of REAL
(- 1) * ((((x - exp_R) `| Z) . x) / ((x - exp_R) . x)) is V22() V23() ext-real Element of REAL
- (exp_R . x) is V22() V23() ext-real Element of REAL
(- (exp_R . x)) / (1 - (exp_R . x)) is V22() V23() ext-real Element of REAL
(- 1) * ((- (exp_R . x)) / (1 - (exp_R . x))) is V22() V23() ext-real Element of REAL
(- 1) * (- (exp_R . x)) is V22() V23() ext-real Element of REAL
((- 1) * (- (exp_R . x))) / (1 - (exp_R . x)) is V22() V23() ext-real Element of REAL
(- 1) (#) (ln * (x - exp_R)) is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
dom ((- 1) (#) (ln * (x - exp_R))) is set
x is V22() V23() ext-real Element of REAL
((- (ln * (x - exp_R))) `| Z) . x is V22() V23() ext-real Element of REAL
exp_R . x is V22() V23() ext-real Element of REAL
1 - (exp_R . x) is V22() V23() ext-real Element of REAL
(exp_R . x) / (1 - (exp_R . x)) is V22() V23() ext-real Element of REAL
(#Z 2) * exp_R is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
Z is open V49() V50() V51() Element of K19(REAL)
x is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
((#Z 2) * exp_R) + x is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
dom (((#Z 2) * exp_R) + x) is set
(((#Z 2) * exp_R) + x) `| Z is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
dom ((#Z 2) * exp_R) is set
dom x is set
(dom ((#Z 2) * exp_R)) /\ (dom x) is set
x is V22() V23() ext-real Element of REAL
x is V22() V23() ext-real Element of REAL
x . x is V22() V23() ext-real Element of REAL
0 * x is V22() V23() ext-real Element of REAL
(0 * x) + 1 is V22() V23() ext-real Element of REAL
x is V22() V23() ext-real Element of REAL
((((#Z 2) * exp_R) + x) `| Z) . x is V22() V23() ext-real Element of REAL
2 * x is V22() V23() ext-real Element of REAL
exp_R (2 * x) is V22() V23() ext-real Element of REAL
2 * (exp_R (2 * x)) is V22() V23() ext-real Element of REAL
diff (((#Z 2) * exp_R),x) is V22() V23() ext-real Element of REAL
diff (x,x) is V22() V23() ext-real Element of REAL
(diff (((#Z 2) * exp_R),x)) + (diff (x,x)) is V22() V23() ext-real Element of REAL
exp_R . x is V22() V23() ext-real Element of REAL
2 - 1 is V22() V23() ext-real V68() Element of REAL
(exp_R . x) #Z (2 - 1) is V22() V23() ext-real Element of REAL
2 * ((exp_R . x) #Z (2 - 1)) is V22() V23() ext-real Element of REAL
diff (exp_R,x) is V22() V23() ext-real Element of REAL
(2 * ((exp_R . x) #Z (2 - 1))) * (diff (exp_R,x)) is V22() V23() ext-real Element of REAL
((2 * ((exp_R . x) #Z (2 - 1))) * (diff (exp_R,x))) + (diff (x,x)) is V22() V23() ext-real Element of REAL
(2 * ((exp_R . x) #Z (2 - 1))) * (exp_R . x) is V22() V23() ext-real Element of REAL
((2 * ((exp_R . x) #Z (2 - 1))) * (exp_R . x)) + (diff (x,x)) is V22() V23() ext-real Element of REAL
2 * (exp_R . x) is V22() V23() ext-real Element of REAL
(2 * (exp_R . x)) * (exp_R . x) is V22() V23() ext-real Element of REAL
((2 * (exp_R . x)) * (exp_R . x)) + (diff (x,x)) is V22() V23() ext-real Element of REAL
(exp_R . x) * (exp_R . x) is V22() V23() ext-real Element of REAL
2 * ((exp_R . x) * (exp_R . x)) is V22() V23() ext-real Element of REAL
(2 * ((exp_R . x) * (exp_R . x))) + (diff (x,x)) is V22() V23() ext-real Element of REAL
exp_R x is V22() V23() ext-real Element of REAL
(exp_R x) * (exp_R . x) is V22() V23() ext-real Element of REAL
2 * ((exp_R x) * (exp_R . x)) is V22() V23() ext-real Element of REAL
(2 * ((exp_R x) * (exp_R . x))) + (diff (x,x)) is V22() V23() ext-real Element of REAL
(exp_R x) * (exp_R x) is V22() V23() ext-real Element of REAL
2 * ((exp_R x) * (exp_R x)) is V22() V23() ext-real Element of REAL
(2 * ((exp_R x) * (exp_R x))) + (diff (x,x)) is V22() V23() ext-real Element of REAL
x + x is V22() V23() ext-real Element of REAL
exp_R (x + x) is V22() V23() ext-real Element of REAL
2 * (exp_R (x + x)) is V22() V23() ext-real Element of REAL
(2 * (exp_R (x + x))) + (diff (x,x)) is V22() V23() ext-real Element of REAL
x `| Z is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
(x `| Z) . x is V22() V23() ext-real Element of REAL
(2 * (exp_R (2 * x))) + ((x `| Z) . x) is V22() V23() ext-real Element of REAL
(2 * (exp_R (2 * x))) + 0 is V22() V23() ext-real Element of REAL
x is V22() V23() ext-real Element of REAL
((((#Z 2) * exp_R) + x) `| Z) . x is V22() V23() ext-real Element of REAL
2 * x is V22() V23() ext-real Element of REAL
exp_R (2 * x) is V22() V23() ext-real Element of REAL
2 * (exp_R (2 * x)) is V22() V23() ext-real Element of REAL
Z is open V49() V50() V51() Element of K19(REAL)
x is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
ln * x is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
(1 / 2) (#) (ln * x) is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
dom ((1 / 2) (#) (ln * x)) is set
((1 / 2) (#) (ln * x)) `| Z is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
x is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
((#Z 2) * exp_R) + x is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
dom (ln * x) is set
dom x is set
f is set
dom (((#Z 2) * exp_R) + x) is set
dom ((#Z 2) * exp_R) is set
dom x is set
(dom ((#Z 2) * exp_R)) /\ (dom x) is set
f is V22() V23() ext-real Element of REAL
x . f is V22() V23() ext-real Element of REAL
exp_R . f is V22() V23() ext-real Element of REAL
(exp_R . f) #Z 2 is V22() V23() ext-real Element of REAL
(((#Z 2) * exp_R) + x) . f is V22() V23() ext-real Element of REAL
((#Z 2) * exp_R) . f is V22() V23() ext-real Element of REAL
x . f is V22() V23() ext-real Element of REAL
(((#Z 2) * exp_R) . f) + (x . f) is V22() V23() ext-real Element of REAL
(#Z 2) . (exp_R . f) is V22() V23() ext-real Element of REAL
((#Z 2) . (exp_R . f)) + (x . f) is V22() V23() ext-real Element of REAL
((exp_R . f) #Z 2) + (x . f) is V22() V23() ext-real Element of REAL
((exp_R . f) #Z 2) + 1 is V22() V23() ext-real Element of REAL
f is V22() V23() ext-real Element of REAL
x . f is V22() V23() ext-real Element of REAL
f is V22() V23() ext-real Element of REAL
(((1 / 2) (#) (ln * x)) `| Z) . f is V22() V23() ext-real Element of REAL
exp_R f is V22() V23() ext-real Element of REAL
- f is V22() V23() ext-real Element of REAL
exp_R (- f) is V22() V23() ext-real Element of REAL
(exp_R f) + (exp_R (- f)) is V22() V23() ext-real Element of REAL
(exp_R f) / ((exp_R f) + (exp_R (- f))) is V22() V23() ext-real Element of REAL
x . f is V22() V23() ext-real Element of REAL
(((#Z 2) * exp_R) + x) . f is V22() V23() ext-real Element of REAL
((#Z 2) * exp_R) . f is V22() V23() ext-real Element of REAL
x . f is V22() V23() ext-real Element of REAL
(((#Z 2) * exp_R) . f) + (x . f) is V22() V23() ext-real Element of REAL
exp_R . f is V22() V23() ext-real Element of REAL
(#Z 2) . (exp_R . f) is V22() V23() ext-real Element of REAL
((#Z 2) . (exp_R . f)) + (x . f) is V22() V23() ext-real Element of REAL
(exp_R . f) #Z 2 is V22() V23() ext-real Element of REAL
((exp_R . f) #Z 2) + (x . f) is V22() V23() ext-real Element of REAL
((exp_R . f) #Z 2) + 1 is V22() V23() ext-real Element of REAL
1 + 1 is V15() V16() V17() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() Element of NAT
(exp_R f) #Z (1 + 1) is V22() V23() ext-real Element of REAL
((exp_R f) #Z (1 + 1)) + 1 is V22() V23() ext-real Element of REAL
(exp_R f) #Z 1 is V22() V23() ext-real Element of REAL
((exp_R f) #Z 1) * ((exp_R f) #Z 1) is V22() V23() ext-real Element of REAL
(((exp_R f) #Z 1) * ((exp_R f) #Z 1)) + 1 is V22() V23() ext-real Element of REAL
(exp_R f) * ((exp_R f) #Z 1) is V22() V23() ext-real Element of REAL
((exp_R f) * ((exp_R f) #Z 1)) + 1 is V22() V23() ext-real Element of REAL
(exp_R f) * (exp_R f) is V22() V23() ext-real Element of REAL
((exp_R f) * (exp_R f)) + 1 is V22() V23() ext-real Element of REAL
diff ((ln * x),f) is V22() V23() ext-real Element of REAL
(1 / 2) * (diff ((ln * x),f)) is V22() V23() ext-real Element of REAL
diff (x,f) is V22() V23() ext-real Element of REAL
(diff (x,f)) / (x . f) is V22() V23() ext-real Element of REAL
(1 / 2) * ((diff (x,f)) / (x . f)) is V22() V23() ext-real Element of REAL
(((#Z 2) * exp_R) + x) `| Z is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
((((#Z 2) * exp_R) + x) `| Z) . f is V22() V23() ext-real Element of REAL
(((((#Z 2) * exp_R) + x) `| Z) . f) / ((((#Z 2) * exp_R) + x) . f) is V22() V23() ext-real Element of REAL
(1 / 2) * ((((((#Z 2) * exp_R) + x) `| Z) . f) / ((((#Z 2) * exp_R) + x) . f)) is V22() V23() ext-real Element of REAL
2 * f is V22() V23() ext-real Element of REAL
exp_R (2 * f) is V22() V23() ext-real Element of REAL
2 * (exp_R (2 * f)) is V22() V23() ext-real Element of REAL
(2 * (exp_R (2 * f))) / (((exp_R f) * (exp_R f)) + 1) is V22() V23() ext-real Element of REAL
(1 / 2) * ((2 * (exp_R (2 * f))) / (((exp_R f) * (exp_R f)) + 1)) is V22() V23() ext-real Element of REAL
(1 / 2) * (2 * (exp_R (2 * f))) is V22() V23() ext-real Element of REAL
((1 / 2) * (2 * (exp_R (2 * f)))) / (((exp_R f) * (exp_R f)) + 1) is V22() V23() ext-real Element of REAL
f + f is V22() V23() ext-real Element of REAL
exp_R (f + f) is V22() V23() ext-real Element of REAL
(exp_R (f + f)) / (exp_R f) is V22() V23() ext-real Element of REAL
(((exp_R f) * (exp_R f)) + 1) / (exp_R f) is V22() V23() ext-real Element of REAL
((exp_R (f + f)) / (exp_R f)) / ((((exp_R f) * (exp_R f)) + 1) / (exp_R f)) is V22() V23() ext-real Element of REAL
((exp_R f) * (exp_R f)) / (exp_R f) is V22() V23() ext-real Element of REAL
(((exp_R f) * (exp_R f)) / (exp_R f)) / ((((exp_R f) * (exp_R f)) + 1) / (exp_R f)) is V22() V23() ext-real Element of REAL
1 / (exp_R f) is V22() V23() ext-real Element of REAL
(((exp_R f) * (exp_R f)) / (exp_R f)) + (1 / (exp_R f)) is V22() V23() ext-real Element of REAL
(((exp_R f) * (exp_R f)) / (exp_R f)) / ((((exp_R f) * (exp_R f)) / (exp_R f)) + (1 / (exp_R f))) is V22() V23() ext-real Element of REAL
(exp_R f) / ((((exp_R f) * (exp_R f)) / (exp_R f)) + (1 / (exp_R f))) is V22() V23() ext-real Element of REAL
(exp_R f) + (1 / (exp_R f)) is V22() V23() ext-real Element of REAL
(exp_R f) / ((exp_R f) + (1 / (exp_R f))) is V22() V23() ext-real Element of REAL
f is V22() V23() ext-real Element of REAL
(((1 / 2) (#) (ln * x)) `| Z) . f is V22() V23() ext-real Element of REAL
exp_R f is V22() V23() ext-real Element of REAL
- f is V22() V23() ext-real Element of REAL
exp_R (- f) is V22() V23() ext-real Element of REAL
(exp_R f) + (exp_R (- f)) is V22() V23() ext-real Element of REAL
(exp_R f) / ((exp_R f) + (exp_R (- f))) is V22() V23() ext-real Element of REAL
Z is open V49() V50() V51() Element of K19(REAL)
x is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
((#Z 2) * exp_R) - x is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
- x is Relation-like V6() V39() set
K58(1) is V22() V23() V68() set
K58(1) (#) x is Relation-like V6() set
((#Z 2) * exp_R) + (- x) is Relation-like V6() set
dom (((#Z 2) * exp_R) - x) is set
(((#Z 2) * exp_R) - x) `| Z is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
dom ((#Z 2) * exp_R) is set
dom x is set
(dom ((#Z 2) * exp_R)) /\ (dom x) is set
x is V22() V23() ext-real Element of REAL
x is V22() V23() ext-real Element of REAL
x . x is V22() V23() ext-real Element of REAL
0 * x is V22() V23() ext-real Element of REAL
(0 * x) + 1 is V22() V23() ext-real Element of REAL
x is V22() V23() ext-real Element of REAL
((((#Z 2) * exp_R) - x) `| Z) . x is V22() V23() ext-real Element of REAL
2 * x is V22() V23() ext-real Element of REAL
exp_R (2 * x) is V22() V23() ext-real Element of REAL
2 * (exp_R (2 * x)) is V22() V23() ext-real Element of REAL
diff (((#Z 2) * exp_R),x) is V22() V23() ext-real Element of REAL
diff (x,x) is V22() V23() ext-real Element of REAL
(diff (((#Z 2) * exp_R),x)) - (diff (x,x)) is V22() V23() ext-real Element of REAL
exp_R . x is V22() V23() ext-real Element of REAL
2 - 1 is V22() V23() ext-real V68() Element of REAL
(exp_R . x) #Z (2 - 1) is V22() V23() ext-real Element of REAL
2 * ((exp_R . x) #Z (2 - 1)) is V22() V23() ext-real Element of REAL
diff (exp_R,x) is V22() V23() ext-real Element of REAL
(2 * ((exp_R . x) #Z (2 - 1))) * (diff (exp_R,x)) is V22() V23() ext-real Element of REAL
((2 * ((exp_R . x) #Z (2 - 1))) * (diff (exp_R,x))) - (diff (x,x)) is V22() V23() ext-real Element of REAL
(exp_R . x) #Z 1 is V22() V23() ext-real Element of REAL
2 * ((exp_R . x) #Z 1) is V22() V23() ext-real Element of REAL
(2 * ((exp_R . x) #Z 1)) * (exp_R . x) is V22() V23() ext-real Element of REAL
((2 * ((exp_R . x) #Z 1)) * (exp_R . x)) - (diff (x,x)) is V22() V23() ext-real Element of REAL
2 * (exp_R . x) is V22() V23() ext-real Element of REAL
(2 * (exp_R . x)) * (exp_R . x) is V22() V23() ext-real Element of REAL
((2 * (exp_R . x)) * (exp_R . x)) - (diff (x,x)) is V22() V23() ext-real Element of REAL
(exp_R . x) * (exp_R . x) is V22() V23() ext-real Element of REAL
2 * ((exp_R . x) * (exp_R . x)) is V22() V23() ext-real Element of REAL
(2 * ((exp_R . x) * (exp_R . x))) - (diff (x,x)) is V22() V23() ext-real Element of REAL
exp_R x is V22() V23() ext-real Element of REAL
(exp_R x) * (exp_R . x) is V22() V23() ext-real Element of REAL
2 * ((exp_R x) * (exp_R . x)) is V22() V23() ext-real Element of REAL
(2 * ((exp_R x) * (exp_R . x))) - (diff (x,x)) is V22() V23() ext-real Element of REAL
(exp_R x) * (exp_R x) is V22() V23() ext-real Element of REAL
2 * ((exp_R x) * (exp_R x)) is V22() V23() ext-real Element of REAL
(2 * ((exp_R x) * (exp_R x))) - (diff (x,x)) is V22() V23() ext-real Element of REAL
x + x is V22() V23() ext-real Element of REAL
exp_R (x + x) is V22() V23() ext-real Element of REAL
2 * (exp_R (x + x)) is V22() V23() ext-real Element of REAL
(2 * (exp_R (x + x))) - (diff (x,x)) is V22() V23() ext-real Element of REAL
x `| Z is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
(x `| Z) . x is V22() V23() ext-real Element of REAL
(2 * (exp_R (2 * x))) - ((x `| Z) . x) is V22() V23() ext-real Element of REAL
(2 * (exp_R (2 * x))) - 0 is V22() V23() ext-real Element of REAL
x is V22() V23() ext-real Element of REAL
((((#Z 2) * exp_R) - x) `| Z) . x is V22() V23() ext-real Element of REAL
2 * x is V22() V23() ext-real Element of REAL
exp_R (2 * x) is V22() V23() ext-real Element of REAL
2 * (exp_R (2 * x)) is V22() V23() ext-real Element of REAL
Z is open V49() V50() V51() Element of K19(REAL)
x is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
ln * x is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
(1 / 2) (#) (ln * x) is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
dom ((1 / 2) (#) (ln * x)) is set
((1 / 2) (#) (ln * x)) `| Z is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
x is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
((#Z 2) * exp_R) - x is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
- x is Relation-like V6() V39() set
K58(1) is V22() V23() V68() set
K58(1) (#) x is Relation-like V6() set
((#Z 2) * exp_R) + (- x) is Relation-like V6() set
dom (ln * x) is set
dom x is set
f is set
dom (((#Z 2) * exp_R) - x) is set
f is V22() V23() ext-real Element of REAL
x . f is V22() V23() ext-real Element of REAL
f is V22() V23() ext-real Element of REAL
x . f is V22() V23() ext-real Element of REAL
dom ((#Z 2) * exp_R) is set
dom x is set
(dom ((#Z 2) * exp_R)) /\ (dom x) is set
f is V22() V23() ext-real Element of REAL
(((1 / 2) (#) (ln * x)) `| Z) . f is V22() V23() ext-real Element of REAL
exp_R f is V22() V23() ext-real Element of REAL
- f is V22() V23() ext-real Element of REAL
exp_R (- f) is V22() V23() ext-real Element of REAL
(exp_R f) - (exp_R (- f)) is V22() V23() ext-real Element of REAL
(exp_R f) / ((exp_R f) - (exp_R (- f))) is V22() V23() ext-real Element of REAL
x . f is V22() V23() ext-real Element of REAL
(((#Z 2) * exp_R) - x) . f is V22() V23() ext-real Element of REAL
((#Z 2) * exp_R) . f is V22() V23() ext-real Element of REAL
x . f is V22() V23() ext-real Element of REAL
(((#Z 2) * exp_R) . f) - (x . f) is V22() V23() ext-real Element of REAL
exp_R . f is V22() V23() ext-real Element of REAL
(#Z 2) . (exp_R . f) is V22() V23() ext-real Element of REAL
((#Z 2) . (exp_R . f)) - (x . f) is V22() V23() ext-real Element of REAL
(exp_R . f) #Z 2 is V22() V23() ext-real Element of REAL
((exp_R . f) #Z 2) - (x . f) is V22() V23() ext-real Element of REAL
((exp_R . f) #Z 2) - 1 is V22() V23() ext-real Element of REAL
1 + 1 is V15() V16() V17() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() Element of NAT
(exp_R f) #Z (1 + 1) is V22() V23() ext-real Element of REAL
((exp_R f) #Z (1 + 1)) - 1 is V22() V23() ext-real Element of REAL
(exp_R f) #Z 1 is V22() V23() ext-real Element of REAL
((exp_R f) #Z 1) * ((exp_R f) #Z 1) is V22() V23() ext-real Element of REAL
(((exp_R f) #Z 1) * ((exp_R f) #Z 1)) - 1 is V22() V23() ext-real Element of REAL
(exp_R f) * ((exp_R f) #Z 1) is V22() V23() ext-real Element of REAL
((exp_R f) * ((exp_R f) #Z 1)) - 1 is V22() V23() ext-real Element of REAL
(exp_R f) * (exp_R f) is V22() V23() ext-real Element of REAL
((exp_R f) * (exp_R f)) - 1 is V22() V23() ext-real Element of REAL
diff ((ln * x),f) is V22() V23() ext-real Element of REAL
(1 / 2) * (diff ((ln * x),f)) is V22() V23() ext-real Element of REAL
diff (x,f) is V22() V23() ext-real Element of REAL
(diff (x,f)) / (x . f) is V22() V23() ext-real Element of REAL
(1 / 2) * ((diff (x,f)) / (x . f)) is V22() V23() ext-real Element of REAL
(((#Z 2) * exp_R) - x) `| Z is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
((((#Z 2) * exp_R) - x) `| Z) . f is V22() V23() ext-real Element of REAL
(((((#Z 2) * exp_R) - x) `| Z) . f) / ((((#Z 2) * exp_R) - x) . f) is V22() V23() ext-real Element of REAL
(1 / 2) * ((((((#Z 2) * exp_R) - x) `| Z) . f) / ((((#Z 2) * exp_R) - x) . f)) is V22() V23() ext-real Element of REAL
2 * f is V22() V23() ext-real Element of REAL
exp_R (2 * f) is V22() V23() ext-real Element of REAL
2 * (exp_R (2 * f)) is V22() V23() ext-real Element of REAL
(2 * (exp_R (2 * f))) / (((exp_R f) * (exp_R f)) - 1) is V22() V23() ext-real Element of REAL
(1 / 2) * ((2 * (exp_R (2 * f))) / (((exp_R f) * (exp_R f)) - 1)) is V22() V23() ext-real Element of REAL
(1 / 2) * (2 * (exp_R (2 * f))) is V22() V23() ext-real Element of REAL
((1 / 2) * (2 * (exp_R (2 * f)))) / (((exp_R f) * (exp_R f)) - 1) is V22() V23() ext-real Element of REAL
f + f is V22() V23() ext-real Element of REAL
exp_R (f + f) is V22() V23() ext-real Element of REAL
(exp_R (f + f)) / (exp_R f) is V22() V23() ext-real Element of REAL
(((exp_R f) * (exp_R f)) - 1) / (exp_R f) is V22() V23() ext-real Element of REAL
((exp_R (f + f)) / (exp_R f)) / ((((exp_R f) * (exp_R f)) - 1) / (exp_R f)) is V22() V23() ext-real Element of REAL
((exp_R f) * (exp_R f)) / (exp_R f) is V22() V23() ext-real Element of REAL
(((exp_R f) * (exp_R f)) / (exp_R f)) / ((((exp_R f) * (exp_R f)) - 1) / (exp_R f)) is V22() V23() ext-real Element of REAL
1 / (exp_R f) is V22() V23() ext-real Element of REAL
(((exp_R f) * (exp_R f)) / (exp_R f)) - (1 / (exp_R f)) is V22() V23() ext-real Element of REAL
(((exp_R f) * (exp_R f)) / (exp_R f)) / ((((exp_R f) * (exp_R f)) / (exp_R f)) - (1 / (exp_R f))) is V22() V23() ext-real Element of REAL
(exp_R f) / ((((exp_R f) * (exp_R f)) / (exp_R f)) - (1 / (exp_R f))) is V22() V23() ext-real Element of REAL
(exp_R f) - (1 / (exp_R f)) is V22() V23() ext-real Element of REAL
(exp_R f) / ((exp_R f) - (1 / (exp_R f))) is V22() V23() ext-real Element of REAL
f is V22() V23() ext-real Element of REAL
(((1 / 2) (#) (ln * x)) `| Z) . f is V22() V23() ext-real Element of REAL
exp_R f is V22() V23() ext-real Element of REAL
- f is V22() V23() ext-real Element of REAL
exp_R (- f) is V22() V23() ext-real Element of REAL
(exp_R f) - (exp_R (- f)) is V22() V23() ext-real Element of REAL
(exp_R f) / ((exp_R f) - (exp_R (- f))) is V22() V23() ext-real Element of REAL
Z is open V49() V50() V51() Element of K19(REAL)
x is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
exp_R - x is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
- x is Relation-like V6() V39() set
K58(1) is V22() V23() V68() set
K58(1) (#) x is Relation-like V6() set
exp_R + (- x) is Relation-like V6() set
(#Z 2) * (exp_R - x) is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
dom ((#Z 2) * (exp_R - x)) is set
((#Z 2) * (exp_R - x)) `| Z is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
dom (exp_R - x) is set
x is set
dom exp_R is set
dom x is set
(dom exp_R) /\ (dom x) is set
x is V22() V23() ext-real Element of REAL
x . x is V22() V23() ext-real Element of REAL
0 * x is V22() V23() ext-real Element of REAL
(0 * x) + 1 is V22() V23() ext-real Element of REAL
x is V22() V23() ext-real Element of REAL
x is V22() V23() ext-real Element of REAL
(((#Z 2) * (exp_R - x)) `| Z) . x is V22() V23() ext-real Element of REAL
exp_R . x is V22() V23() ext-real Element of REAL
2 * (exp_R . x) is V22() V23() ext-real Element of REAL
(exp_R . x) - 1 is V22() V23() ext-real Element of REAL
(2 * (exp_R . x)) * ((exp_R . x) - 1) is V22() V23() ext-real Element of REAL
(exp_R - x) . x is V22() V23() ext-real Element of REAL
x . x is V22() V23() ext-real Element of REAL
(exp_R . x) - (x . x) is V22() V23() ext-real Element of REAL
diff ((exp_R - x),x) is V22() V23() ext-real Element of REAL
diff (exp_R,x) is V22() V23() ext-real Element of REAL
diff (x,x) is V22() V23() ext-real Element of REAL
(diff (exp_R,x)) - (diff (x,x)) is V22() V23() ext-real Element of REAL
x `| Z is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
(x `| Z) . x is V22() V23() ext-real Element of REAL
(diff (exp_R,x)) - ((x `| Z) . x) is V22() V23() ext-real Element of REAL
(exp_R . x) - ((x `| Z) . x) is V22() V23() ext-real Element of REAL
(exp_R . x) - 0 is V22() V23() ext-real Element of REAL
diff (((#Z 2) * (exp_R - x)),x) is V22() V23() ext-real Element of REAL
2 - 1 is V22() V23() ext-real V68() Element of REAL
((exp_R - x) . x) #Z (2 - 1) is V22() V23() ext-real Element of REAL
2 * (((exp_R - x) . x) #Z (2 - 1)) is V22() V23() ext-real Element of REAL
(2 * (((exp_R - x) . x) #Z (2 - 1))) * (diff ((exp_R - x),x)) is V22() V23() ext-real Element of REAL
2 * ((exp_R . x) - 1) is V22() V23() ext-real Element of REAL
(2 * ((exp_R . x) - 1)) * (exp_R . x) is V22() V23() ext-real Element of REAL
x is V22() V23() ext-real Element of REAL
(((#Z 2) * (exp_R - x)) `| Z) . x is V22() V23() ext-real Element of REAL
exp_R . x is V22() V23() ext-real Element of REAL
2 * (exp_R . x) is V22() V23() ext-real Element of REAL
(exp_R . x) - 1 is V22() V23() ext-real Element of REAL
(2 * (exp_R . x)) * ((exp_R . x) - 1) is V22() V23() ext-real Element of REAL
Z is open V49() V50() V51() Element of K19(REAL)
x is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
dom x is set
x `| Z is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
x is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
exp_R - x is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
- x is Relation-like V6() V39() set
K58(1) is V22() V23() V68() set
K58(1) (#) x is Relation-like V6() set
exp_R + (- x) is Relation-like V6() set
(#Z 2) * (exp_R - x) is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
((#Z 2) * (exp_R - x)) / exp_R is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
ln * (((#Z 2) * (exp_R - x)) / exp_R) is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
dom (((#Z 2) * (exp_R - x)) / exp_R) is set
f is set
dom exp_R is set
dom ((#Z 2) * (exp_R - x)) is set
exp_R " {0} is set
(dom exp_R) \ (exp_R " {0}) is Element of K19((dom exp_R))
K19((dom exp_R)) is set
(dom ((#Z 2) * (exp_R - x))) /\ ((dom exp_R) \ (exp_R " {0})) is Element of K19((dom exp_R))
dom (exp_R - x) is set
f is set
f is V22() V23() ext-real Element of REAL
(((#Z 2) * (exp_R - x)) / exp_R) . f is V22() V23() ext-real Element of REAL
exp_R . f is V22() V23() ext-real Element of REAL
(exp_R - x) . f is V22() V23() ext-real Element of REAL
((exp_R - x) . f) #Z 2 is V22() V23() ext-real Element of REAL
((#Z 2) * (exp_R - x)) . f is V22() V23() ext-real Element of REAL
(exp_R . f) " is V22() V23() ext-real Element of REAL
(((#Z 2) * (exp_R - x)) . f) * ((exp_R . f) ") is V22() V23() ext-real Element of REAL
1 / (exp_R . f) is V22() V23() ext-real Element of REAL
(((#Z 2) * (exp_R - x)) . f) * (1 / (exp_R . f)) is V22() V23() ext-real Element of REAL
(((#Z 2) * (exp_R - x)) . f) / (exp_R . f) is V22() V23() ext-real Element of REAL
(#Z 2) . ((exp_R - x) . f) is V22() V23() ext-real Element of REAL
((#Z 2) . ((exp_R - x) . f)) / (exp_R . f) is V22() V23() ext-real Element of REAL
(((exp_R - x) . f) #Z 2) / (exp_R . f) is V22() V23() ext-real Element of REAL
f is V22() V23() ext-real Element of REAL
x . f is V22() V23() ext-real Element of REAL
f is V22() V23() ext-real Element of REAL
exp_R . f is V22() V23() ext-real Element of REAL
f is V22() V23() ext-real Element of REAL
(((#Z 2) * (exp_R - x)) / exp_R) . f is V22() V23() ext-real Element of REAL
f is V22() V23() ext-real Element of REAL
(x `| Z) . f is V22() V23() ext-real Element of REAL
exp_R . f is V22() V23() ext-real Element of REAL
(exp_R . f) + 1 is V22() V23() ext-real Element of REAL
(exp_R . f) - 1 is V22() V23() ext-real Element of REAL
((exp_R . f) + 1) / ((exp_R . f) - 1) is V22() V23() ext-real Element of REAL
(exp_R - x) . f is V22() V23() ext-real Element of REAL
x . f is V22() V23() ext-real Element of REAL
(exp_R . f) - (x . f) is V22() V23() ext-real Element of REAL
(((#Z 2) * (exp_R - x)) / exp_R) . f is V22() V23() ext-real Element of REAL
((#Z 2) * (exp_R - x)) . f is V22() V23() ext-real Element of REAL
(exp_R . f) " is V22() V23() ext-real Element of REAL
(((#Z 2) * (exp_R - x)) . f) * ((exp_R . f) ") is V22() V23() ext-real Element of REAL
1 / (exp_R . f) is V22() V23() ext-real Element of REAL
(((#Z 2) * (exp_R - x)) . f) * (1 / (exp_R . f)) is V22() V23() ext-real Element of REAL
(((#Z 2) * (exp_R - x)) . f) / (exp_R . f) is V22() V23() ext-real Element of REAL
(#Z 2) . ((exp_R - x) . f) is V22() V23() ext-real Element of REAL
((#Z 2) . ((exp_R - x) . f)) / (exp_R . f) is V22() V23() ext-real Element of REAL
1 + 1 is V15() V16() V17() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() Element of NAT
((exp_R . f) - 1) #Z (1 + 1) is V22() V23() ext-real Element of REAL
(((exp_R . f) - 1) #Z (1 + 1)) / (exp_R . f) is V22() V23() ext-real Element of REAL
((exp_R . f) - 1) #Z 1 is V22() V23() ext-real Element of REAL
(((exp_R . f) - 1) #Z 1) * (((exp_R . f) - 1) #Z 1) is V22() V23() ext-real Element of REAL
((((exp_R . f) - 1) #Z 1) * (((exp_R . f) - 1) #Z 1)) / (exp_R . f) is V22() V23() ext-real Element of REAL
((exp_R . f) - 1) * (((exp_R . f) - 1) #Z 1) is V22() V23() ext-real Element of REAL
(((exp_R . f) - 1) * (((exp_R . f) - 1) #Z 1)) / (exp_R . f) is V22() V23() ext-real Element of REAL
((exp_R . f) - 1) * ((exp_R . f) - 1) is V22() V23() ext-real Element of REAL
(((exp_R . f) - 1) * ((exp_R . f) - 1)) / (exp_R . f) is V22() V23() ext-real Element of REAL
diff ((((#Z 2) * (exp_R - x)) / exp_R),f) is V22() V23() ext-real Element of REAL
diff (((#Z 2) * (exp_R - x)),f) is V22() V23() ext-real Element of REAL
(diff (((#Z 2) * (exp_R - x)),f)) * (exp_R . f) is V22() V23() ext-real Element of REAL
diff (exp_R,f) is V22() V23() ext-real Element of REAL
(diff (exp_R,f)) * (((#Z 2) * (exp_R - x)) . f) is V22() V23() ext-real Element of REAL
((diff (((#Z 2) * (exp_R - x)),f)) * (exp_R . f)) - ((diff (exp_R,f)) * (((#Z 2) * (exp_R - x)) . f)) is V22() V23() ext-real Element of REAL
(exp_R . f) ^2 is V22() V23() ext-real Element of REAL
K57((exp_R . f),(exp_R . f)) is set
(((diff (((#Z 2) * (exp_R - x)),f)) * (exp_R . f)) - ((diff (exp_R,f)) * (((#Z 2) * (exp_R - x)) . f))) / ((exp_R . f) ^2) is V22() V23() ext-real Element of REAL
((#Z 2) * (exp_R - x)) `| Z is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
(((#Z 2) * (exp_R - x)) `| Z) . f is V22() V23() ext-real Element of REAL
((((#Z 2) * (exp_R - x)) `| Z) . f) * (exp_R . f) is V22() V23() ext-real Element of REAL
(((((#Z 2) * (exp_R - x)) `| Z) . f) * (exp_R . f)) - ((diff (exp_R,f)) * (((#Z 2) * (exp_R - x)) . f)) is V22() V23() ext-real Element of REAL
((((((#Z 2) * (exp_R - x)) `| Z) . f) * (exp_R . f)) - ((diff (exp_R,f)) * (((#Z 2) * (exp_R - x)) . f))) / ((exp_R . f) ^2) is V22() V23() ext-real Element of REAL
2 * (exp_R . f) is V22() V23() ext-real Element of REAL
(2 * (exp_R . f)) * ((exp_R . f) - 1) is V22() V23() ext-real Element of REAL
((2 * (exp_R . f)) * ((exp_R . f) - 1)) * (exp_R . f) is V22() V23() ext-real Element of REAL
(((2 * (exp_R . f)) * ((exp_R . f) - 1)) * (exp_R . f)) - ((diff (exp_R,f)) * (((#Z 2) * (exp_R - x)) . f)) is V22() V23() ext-real Element of REAL
((((2 * (exp_R . f)) * ((exp_R . f) - 1)) * (exp_R . f)) - ((diff (exp_R,f)) * (((#Z 2) * (exp_R - x)) . f))) / ((exp_R . f) ^2) is V22() V23() ext-real Element of REAL
(exp_R . f) * (((#Z 2) * (exp_R - x)) . f) is V22() V23() ext-real Element of REAL
(((2 * (exp_R . f)) * ((exp_R . f) - 1)) * (exp_R . f)) - ((exp_R . f) * (((#Z 2) * (exp_R - x)) . f)) is V22() V23() ext-real Element of REAL
((((2 * (exp_R . f)) * ((exp_R . f) - 1)) * (exp_R . f)) - ((exp_R . f) * (((#Z 2) * (exp_R - x)) . f))) / ((exp_R . f) ^2) is V22() V23() ext-real Element of REAL
((2 * (exp_R . f)) * ((exp_R . f) - 1)) - (((#Z 2) * (exp_R - x)) . f) is V22() V23() ext-real Element of REAL
(((2 * (exp_R . f)) * ((exp_R . f) - 1)) - (((#Z 2) * (exp_R - x)) . f)) * (exp_R . f) is V22() V23() ext-real Element of REAL
(exp_R . f) * (exp_R . f) is V22() V23() ext-real Element of REAL
((((2 * (exp_R . f)) * ((exp_R . f) - 1)) - (((#Z 2) * (exp_R - x)) . f)) * (exp_R . f)) / ((exp_R . f) * (exp_R . f)) is V22() V23() ext-real Element of REAL
(((2 * (exp_R . f)) * ((exp_R . f) - 1)) - (((#Z 2) * (exp_R - x)) . f)) / (exp_R . f) is V22() V23() ext-real Element of REAL
((2 * (exp_R . f)) * ((exp_R . f) - 1)) - ((#Z 2) . ((exp_R - x) . f)) is V22() V23() ext-real Element of REAL
(((2 * (exp_R . f)) * ((exp_R . f) - 1)) - ((#Z 2) . ((exp_R - x) . f))) / (exp_R . f) is V22() V23() ext-real Element of REAL
((exp_R - x) . f) #Z 2 is V22() V23() ext-real Element of REAL
((2 * (exp_R . f)) * ((exp_R . f) - 1)) - (((exp_R - x) . f) #Z 2) is V22() V23() ext-real Element of REAL
(((2 * (exp_R . f)) * ((exp_R . f) - 1)) - (((exp_R - x) . f) #Z 2)) / (exp_R . f) is V22() V23() ext-real Element of REAL
((exp_R . f) - (x . f)) #Z 2 is V22() V23() ext-real Element of REAL
((2 * (exp_R . f)) * ((exp_R . f) - 1)) - (((exp_R . f) - (x . f)) #Z 2) is V22() V23() ext-real Element of REAL
(((2 * (exp_R . f)) * ((exp_R . f) - 1)) - (((exp_R . f) - (x . f)) #Z 2)) / (exp_R . f) is V22() V23() ext-real Element of REAL
((2 * (exp_R . f)) * ((exp_R . f) - 1)) - (((exp_R . f) - 1) #Z (1 + 1)) is V22() V23() ext-real Element of REAL
(((2 * (exp_R . f)) * ((exp_R . f) - 1)) - (((exp_R . f) - 1) #Z (1 + 1))) / (exp_R . f) is V22() V23() ext-real Element of REAL
((2 * (exp_R . f)) * ((exp_R . f) - 1)) - ((((exp_R . f) - 1) #Z 1) * (((exp_R . f) - 1) #Z 1)) is V22() V23() ext-real Element of REAL
(((2 * (exp_R . f)) * ((exp_R . f) - 1)) - ((((exp_R . f) - 1) #Z 1) * (((exp_R . f) - 1) #Z 1))) / (exp_R . f) is V22() V23() ext-real Element of REAL
((2 * (exp_R . f)) * ((exp_R . f) - 1)) - (((exp_R . f) - 1) * (((exp_R . f) - 1) #Z 1)) is V22() V23() ext-real Element of REAL
(((2 * (exp_R . f)) * ((exp_R . f) - 1)) - (((exp_R . f) - 1) * (((exp_R . f) - 1) #Z 1))) / (exp_R . f) is V22() V23() ext-real Element of REAL
((2 * (exp_R . f)) * ((exp_R . f) - 1)) - (((exp_R . f) - 1) * ((exp_R . f) - 1)) is V22() V23() ext-real Element of REAL
(((2 * (exp_R . f)) * ((exp_R . f) - 1)) - (((exp_R . f) - 1) * ((exp_R . f) - 1))) / (exp_R . f) is V22() V23() ext-real Element of REAL
((exp_R . f) + 1) * ((exp_R . f) - 1) is V22() V23() ext-real Element of REAL
(((exp_R . f) + 1) * ((exp_R . f) - 1)) / (exp_R . f) is V22() V23() ext-real Element of REAL
diff ((ln * (((#Z 2) * (exp_R - x)) / exp_R)),f) is V22() V23() ext-real Element of REAL
((((exp_R . f) + 1) * ((exp_R . f) - 1)) / (exp_R . f)) / ((((exp_R . f) - 1) * ((exp_R . f) - 1)) / (exp_R . f)) is V22() V23() ext-real Element of REAL
(((exp_R . f) + 1) * ((exp_R . f) - 1)) / (((exp_R . f) - 1) * ((exp_R . f) - 1)) is V22() V23() ext-real Element of REAL
f is V22() V23() ext-real Element of REAL
(x `| Z) . f is V22() V23() ext-real Element of REAL
exp_R . f is V22() V23() ext-real Element of REAL
(exp_R . f) + 1 is V22() V23() ext-real Element of REAL
(exp_R . f) - 1 is V22() V23() ext-real Element of REAL
((exp_R . f) + 1) / ((exp_R . f) - 1) is V22() V23() ext-real Element of REAL
Z is open V49() V50() V51() Element of K19(REAL)
x is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
exp_R + x is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
(#Z 2) * (exp_R + x) is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
dom ((#Z 2) * (exp_R + x)) is set
((#Z 2) * (exp_R + x)) `| Z is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
dom (exp_R + x) is set
x is set
dom exp_R is set
dom x is set
(dom exp_R) /\ (dom x) is set
x is V22() V23() ext-real Element of REAL
x . x is V22() V23() ext-real Element of REAL
0 * x is V22() V23() ext-real Element of REAL
(0 * x) + 1 is V22() V23() ext-real Element of REAL
x is V22() V23() ext-real Element of REAL
x is V22() V23() ext-real Element of REAL
(((#Z 2) * (exp_R + x)) `| Z) . x is V22() V23() ext-real Element of REAL
exp_R . x is V22() V23() ext-real Element of REAL
2 * (exp_R . x) is V22() V23() ext-real Element of REAL
(exp_R . x) + 1 is V22() V23() ext-real Element of REAL
(2 * (exp_R . x)) * ((exp_R . x) + 1) is V22() V23() ext-real Element of REAL
(exp_R + x) . x is V22() V23() ext-real Element of REAL
x . x is V22() V23() ext-real Element of REAL
(exp_R . x) + (x . x) is V22() V23() ext-real Element of REAL
diff ((exp_R + x),x) is V22() V23() ext-real Element of REAL
diff (exp_R,x) is V22() V23() ext-real Element of REAL
diff (x,x) is V22() V23() ext-real Element of REAL
(diff (exp_R,x)) + (diff (x,x)) is V22() V23() ext-real Element of REAL
x `| Z is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
(x `| Z) . x is V22() V23() ext-real Element of REAL
(diff (exp_R,x)) + ((x `| Z) . x) is V22() V23() ext-real Element of REAL
(exp_R . x) + ((x `| Z) . x) is V22() V23() ext-real Element of REAL
(exp_R . x) + 0 is V22() V23() ext-real Element of REAL
diff (((#Z 2) * (exp_R + x)),x) is V22() V23() ext-real Element of REAL
2 - 1 is V22() V23() ext-real V68() Element of REAL
((exp_R + x) . x) #Z (2 - 1) is V22() V23() ext-real Element of REAL
2 * (((exp_R + x) . x) #Z (2 - 1)) is V22() V23() ext-real Element of REAL
(2 * (((exp_R + x) . x) #Z (2 - 1))) * (diff ((exp_R + x),x)) is V22() V23() ext-real Element of REAL
2 * ((exp_R . x) + 1) is V22() V23() ext-real Element of REAL
(2 * ((exp_R . x) + 1)) * (exp_R . x) is V22() V23() ext-real Element of REAL
x is V22() V23() ext-real Element of REAL
(((#Z 2) * (exp_R + x)) `| Z) . x is V22() V23() ext-real Element of REAL
exp_R . x is V22() V23() ext-real Element of REAL
2 * (exp_R . x) is V22() V23() ext-real Element of REAL
(exp_R . x) + 1 is V22() V23() ext-real Element of REAL
(2 * (exp_R . x)) * ((exp_R . x) + 1) is V22() V23() ext-real Element of REAL
Z is open V49() V50() V51() Element of K19(REAL)
x is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
dom x is set
x `| Z is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
x is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
exp_R + x is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
(#Z 2) * (exp_R + x) is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
((#Z 2) * (exp_R + x)) / exp_R is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
ln * (((#Z 2) * (exp_R + x)) / exp_R) is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
dom (((#Z 2) * (exp_R + x)) / exp_R) is set
f is set
dom exp_R is set
dom ((#Z 2) * (exp_R + x)) is set
exp_R " {0} is set
(dom exp_R) \ (exp_R " {0}) is Element of K19((dom exp_R))
K19((dom exp_R)) is set
(dom ((#Z 2) * (exp_R + x))) /\ ((dom exp_R) \ (exp_R " {0})) is Element of K19((dom exp_R))
dom (exp_R + x) is set
f is set
f is V22() V23() ext-real Element of REAL
(((#Z 2) * (exp_R + x)) / exp_R) . f is V22() V23() ext-real Element of REAL
exp_R . f is V22() V23() ext-real Element of REAL
(exp_R + x) . f is V22() V23() ext-real Element of REAL
x . f is V22() V23() ext-real Element of REAL
(exp_R . f) + (x . f) is V22() V23() ext-real Element of REAL
(exp_R . f) + 1 is V22() V23() ext-real Element of REAL
((exp_R + x) . f) #Z 2 is V22() V23() ext-real Element of REAL
((#Z 2) * (exp_R + x)) . f is V22() V23() ext-real Element of REAL
(exp_R . f) " is V22() V23() ext-real Element of REAL
(((#Z 2) * (exp_R + x)) . f) * ((exp_R . f) ") is V22() V23() ext-real Element of REAL
1 / (exp_R . f) is V22() V23() ext-real Element of REAL
(((#Z 2) * (exp_R + x)) . f) * (1 / (exp_R . f)) is V22() V23() ext-real Element of REAL
(((#Z 2) * (exp_R + x)) . f) / (exp_R . f) is V22() V23() ext-real Element of REAL
(#Z 2) . ((exp_R + x) . f) is V22() V23() ext-real Element of REAL
((#Z 2) . ((exp_R + x) . f)) / (exp_R . f) is V22() V23() ext-real Element of REAL
(((exp_R + x) . f) #Z 2) / (exp_R . f) is V22() V23() ext-real Element of REAL
f is V22() V23() ext-real Element of REAL
exp_R . f is V22() V23() ext-real Element of REAL
f is V22() V23() ext-real Element of REAL
(((#Z 2) * (exp_R + x)) / exp_R) . f is V22() V23() ext-real Element of REAL
f is V22() V23() ext-real Element of REAL
(x `| Z) . f is V22() V23() ext-real Element of REAL
exp_R . f is V22() V23() ext-real Element of REAL
(exp_R . f) - 1 is V22() V23() ext-real Element of REAL
(exp_R . f) + 1 is V22() V23() ext-real Element of REAL
((exp_R . f) - 1) / ((exp_R . f) + 1) is V22() V23() ext-real Element of REAL
(exp_R + x) . f is V22() V23() ext-real Element of REAL
x . f is V22() V23() ext-real Element of REAL
(exp_R . f) + (x . f) is V22() V23() ext-real Element of REAL
((#Z 2) * (exp_R + x)) . f is V22() V23() ext-real Element of REAL
(#Z 2) . ((exp_R + x) . f) is V22() V23() ext-real Element of REAL
1 + 1 is V15() V16() V17() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() Element of NAT
((exp_R . f) + 1) #Z (1 + 1) is V22() V23() ext-real Element of REAL
((exp_R . f) + 1) #Z 1 is V22() V23() ext-real Element of REAL
(((exp_R . f) + 1) #Z 1) * (((exp_R . f) + 1) #Z 1) is V22() V23() ext-real Element of REAL
((exp_R . f) + 1) * (((exp_R . f) + 1) #Z 1) is V22() V23() ext-real Element of REAL
((exp_R . f) + 1) * ((exp_R . f) + 1) is V22() V23() ext-real Element of REAL
(((#Z 2) * (exp_R + x)) / exp_R) . f is V22() V23() ext-real Element of REAL
diff ((((#Z 2) * (exp_R + x)) / exp_R),f) is V22() V23() ext-real Element of REAL
diff (((#Z 2) * (exp_R + x)),f) is V22() V23() ext-real Element of REAL
(diff (((#Z 2) * (exp_R + x)),f)) * (exp_R . f) is V22() V23() ext-real Element of REAL
diff (exp_R,f) is V22() V23() ext-real Element of REAL
(diff (exp_R,f)) * (((#Z 2) * (exp_R + x)) . f) is V22() V23() ext-real Element of REAL
((diff (((#Z 2) * (exp_R + x)),f)) * (exp_R . f)) - ((diff (exp_R,f)) * (((#Z 2) * (exp_R + x)) . f)) is V22() V23() ext-real Element of REAL
(exp_R . f) ^2 is V22() V23() ext-real Element of REAL
K57((exp_R . f),(exp_R . f)) is set
(((diff (((#Z 2) * (exp_R + x)),f)) * (exp_R . f)) - ((diff (exp_R,f)) * (((#Z 2) * (exp_R + x)) . f))) / ((exp_R . f) ^2) is V22() V23() ext-real Element of REAL
((#Z 2) * (exp_R + x)) `| Z is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
(((#Z 2) * (exp_R + x)) `| Z) . f is V22() V23() ext-real Element of REAL
((((#Z 2) * (exp_R + x)) `| Z) . f) * (exp_R . f) is V22() V23() ext-real Element of REAL
(((((#Z 2) * (exp_R + x)) `| Z) . f) * (exp_R . f)) - ((diff (exp_R,f)) * (((#Z 2) * (exp_R + x)) . f)) is V22() V23() ext-real Element of REAL
((((((#Z 2) * (exp_R + x)) `| Z) . f) * (exp_R . f)) - ((diff (exp_R,f)) * (((#Z 2) * (exp_R + x)) . f))) / ((exp_R . f) ^2) is V22() V23() ext-real Element of REAL
2 * (exp_R . f) is V22() V23() ext-real Element of REAL
(2 * (exp_R . f)) * ((exp_R . f) + 1) is V22() V23() ext-real Element of REAL
((2 * (exp_R . f)) * ((exp_R . f) + 1)) * (exp_R . f) is V22() V23() ext-real Element of REAL
(((2 * (exp_R . f)) * ((exp_R . f) + 1)) * (exp_R . f)) - ((diff (exp_R,f)) * (((#Z 2) * (exp_R + x)) . f)) is V22() V23() ext-real Element of REAL
((((2 * (exp_R . f)) * ((exp_R . f) + 1)) * (exp_R . f)) - ((diff (exp_R,f)) * (((#Z 2) * (exp_R + x)) . f))) / ((exp_R . f) ^2) is V22() V23() ext-real Element of REAL
(exp_R . f) * (((#Z 2) * (exp_R + x)) . f) is V22() V23() ext-real Element of REAL
(((2 * (exp_R . f)) * ((exp_R . f) + 1)) * (exp_R . f)) - ((exp_R . f) * (((#Z 2) * (exp_R + x)) . f)) is V22() V23() ext-real Element of REAL
((((2 * (exp_R . f)) * ((exp_R . f) + 1)) * (exp_R . f)) - ((exp_R . f) * (((#Z 2) * (exp_R + x)) . f))) / ((exp_R . f) ^2) is V22() V23() ext-real Element of REAL
((2 * (exp_R . f)) * ((exp_R . f) + 1)) - (((#Z 2) * (exp_R + x)) . f) is V22() V23() ext-real Element of REAL
(((2 * (exp_R . f)) * ((exp_R . f) + 1)) - (((#Z 2) * (exp_R + x)) . f)) * (exp_R . f) is V22() V23() ext-real Element of REAL
(exp_R . f) * (exp_R . f) is V22() V23() ext-real Element of REAL
((((2 * (exp_R . f)) * ((exp_R . f) + 1)) - (((#Z 2) * (exp_R + x)) . f)) * (exp_R . f)) / ((exp_R . f) * (exp_R . f)) is V22() V23() ext-real Element of REAL
((exp_R . f) - 1) * ((exp_R . f) + 1) is V22() V23() ext-real Element of REAL
(((exp_R . f) - 1) * ((exp_R . f) + 1)) / (exp_R . f) is V22() V23() ext-real Element of REAL
(exp_R . f) " is V22() V23() ext-real Element of REAL
(((#Z 2) * (exp_R + x)) . f) * ((exp_R . f) ") is V22() V23() ext-real Element of REAL
1 / (exp_R . f) is V22() V23() ext-real Element of REAL
(((#Z 2) * (exp_R + x)) . f) * (1 / (exp_R . f)) is V22() V23() ext-real Element of REAL
(((exp_R . f) + 1) * ((exp_R . f) + 1)) / (exp_R . f) is V22() V23() ext-real Element of REAL
diff ((ln * (((#Z 2) * (exp_R + x)) / exp_R)),f) is V22() V23() ext-real Element of REAL
((exp_R . f) + 1) * ((exp_R . f) - 1) is V22() V23() ext-real Element of REAL
(((exp_R . f) + 1) * ((exp_R . f) - 1)) / (exp_R . f) is V22() V23() ext-real Element of REAL
((((exp_R . f) + 1) * ((exp_R . f) - 1)) / (exp_R . f)) / ((((exp_R . f) + 1) * ((exp_R . f) + 1)) / (exp_R . f)) is V22() V23() ext-real Element of REAL
(((exp_R . f) + 1) * ((exp_R . f) - 1)) / (((exp_R . f) + 1) * ((exp_R . f) + 1)) is V22() V23() ext-real Element of REAL
f is V22() V23() ext-real Element of REAL
(x `| Z) . f is V22() V23() ext-real Element of REAL
exp_R . f is V22() V23() ext-real Element of REAL
(exp_R . f) - 1 is V22() V23() ext-real Element of REAL
(exp_R . f) + 1 is V22() V23() ext-real Element of REAL
((exp_R . f) - 1) / ((exp_R . f) + 1) is V22() V23() ext-real Element of REAL
Z is open V49() V50() V51() Element of K19(REAL)
x is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
x - exp_R is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
- exp_R is Relation-like V6() V39() set
K58(1) is V22() V23() V68() set
K58(1) (#) exp_R is Relation-like V6() set
x + (- exp_R) is Relation-like V6() set
(#Z 2) * (x - exp_R) is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
dom ((#Z 2) * (x - exp_R)) is set
((#Z 2) * (x - exp_R)) `| Z is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
dom (x - exp_R) is set
x is set
dom exp_R is set
dom x is set
(dom exp_R) /\ (dom x) is set
x is V22() V23() ext-real Element of REAL
x . x is V22() V23() ext-real Element of REAL
0 * x is V22() V23() ext-real Element of REAL
(0 * x) + 1 is V22() V23() ext-real Element of REAL
x is V22() V23() ext-real Element of REAL
x is V22() V23() ext-real Element of REAL
(((#Z 2) * (x - exp_R)) `| Z) . x is V22() V23() ext-real Element of REAL
exp_R . x is V22() V23() ext-real Element of REAL
2 * (exp_R . x) is V22() V23() ext-real Element of REAL
1 - (exp_R . x) is V22() V23() ext-real Element of REAL
(2 * (exp_R . x)) * (1 - (exp_R . x)) is V22() V23() ext-real Element of REAL
- ((2 * (exp_R . x)) * (1 - (exp_R . x))) is V22() V23() ext-real Element of REAL
(x - exp_R) . x is V22() V23() ext-real Element of REAL
x . x is V22() V23() ext-real Element of REAL
(x . x) - (exp_R . x) is V22() V23() ext-real Element of REAL
diff ((x - exp_R),x) is V22() V23() ext-real Element of REAL
diff (x,x) is V22() V23() ext-real Element of REAL
diff (exp_R,x) is V22() V23() ext-real Element of REAL
(diff (x,x)) - (diff (exp_R,x)) is V22() V23() ext-real Element of REAL
x `| Z is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
(x `| Z) . x is V22() V23() ext-real Element of REAL
((x `| Z) . x) - (diff (exp_R,x)) is V22() V23() ext-real Element of REAL
((x `| Z) . x) - (exp_R . x) is V22() V23() ext-real Element of REAL
0 - (exp_R . x) is V22() V23() ext-real Element of REAL
- (exp_R . x) is V22() V23() ext-real Element of REAL
diff (((#Z 2) * (x - exp_R)),x) is V22() V23() ext-real Element of REAL
2 - 1 is V22() V23() ext-real V68() Element of REAL
((x - exp_R) . x) #Z (2 - 1) is V22() V23() ext-real Element of REAL
2 * (((x - exp_R) . x) #Z (2 - 1)) is V22() V23() ext-real Element of REAL
(2 * (((x - exp_R) . x) #Z (2 - 1))) * (diff ((x - exp_R),x)) is V22() V23() ext-real Element of REAL
2 * (1 - (exp_R . x)) is V22() V23() ext-real Element of REAL
(2 * (1 - (exp_R . x))) * (- (exp_R . x)) is V22() V23() ext-real Element of REAL
x is V22() V23() ext-real Element of REAL
(((#Z 2) * (x - exp_R)) `| Z) . x is V22() V23() ext-real Element of REAL
exp_R . x is V22() V23() ext-real Element of REAL
2 * (exp_R . x) is V22() V23() ext-real Element of REAL
1 - (exp_R . x) is V22() V23() ext-real Element of REAL
(2 * (exp_R . x)) * (1 - (exp_R . x)) is V22() V23() ext-real Element of REAL
- ((2 * (exp_R . x)) * (1 - (exp_R . x))) is V22() V23() ext-real Element of REAL
Z is open V49() V50() V51() Element of K19(REAL)
x is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
dom x is set
x `| Z is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
x is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
x - exp_R is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
- exp_R is Relation-like V6() V39() set
K58(1) is V22() V23() V68() set
K58(1) (#) exp_R is Relation-like V6() set
x + (- exp_R) is Relation-like V6() set
(#Z 2) * (x - exp_R) is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
exp_R / ((#Z 2) * (x - exp_R)) is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
ln * (exp_R / ((#Z 2) * (x - exp_R))) is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
dom (exp_R / ((#Z 2) * (x - exp_R))) is set
f is set
dom ((#Z 2) * (x - exp_R)) is set
dom exp_R is set
((#Z 2) * (x - exp_R)) " {0} is set
(dom ((#Z 2) * (x - exp_R))) \ (((#Z 2) * (x - exp_R)) " {0}) is Element of K19((dom ((#Z 2) * (x - exp_R))))
K19((dom ((#Z 2) * (x - exp_R)))) is set
(dom exp_R) /\ ((dom ((#Z 2) * (x - exp_R))) \ (((#Z 2) * (x - exp_R)) " {0})) is Element of K19((dom ((#Z 2) * (x - exp_R))))
dom (x - exp_R) is set
f is set
f is V22() V23() ext-real Element of REAL
((#Z 2) * (x - exp_R)) . f is V22() V23() ext-real Element of REAL
(x - exp_R) . f is V22() V23() ext-real Element of REAL
(#Z 2) . ((x - exp_R) . f) is V22() V23() ext-real Element of REAL
((x - exp_R) . f) #Z 2 is V22() V23() ext-real Element of REAL
f is V22() V23() ext-real Element of REAL
(exp_R / ((#Z 2) * (x - exp_R))) . f is V22() V23() ext-real Element of REAL
exp_R . f is V22() V23() ext-real Element of REAL
((#Z 2) * (x - exp_R)) . f is V22() V23() ext-real Element of REAL
(((#Z 2) * (x - exp_R)) . f) " is V22() V23() ext-real Element of REAL
(exp_R . f) * ((((#Z 2) * (x - exp_R)) . f) ") is V22() V23() ext-real Element of REAL
1 / (((#Z 2) * (x - exp_R)) . f) is V22() V23() ext-real Element of REAL
(exp_R . f) * (1 / (((#Z 2) * (x - exp_R)) . f)) is V22() V23() ext-real Element of REAL
(exp_R . f) / (((#Z 2) * (x - exp_R)) . f) is V22() V23() ext-real Element of REAL
f is V22() V23() ext-real Element of REAL
x . f is V22() V23() ext-real Element of REAL
f is V22() V23() ext-real Element of REAL
((#Z 2) * (x - exp_R)) . f is V22() V23() ext-real Element of REAL
f is V22() V23() ext-real Element of REAL
(exp_R / ((#Z 2) * (x - exp_R))) . f is V22() V23() ext-real Element of REAL
f is V22() V23() ext-real Element of REAL
(x `| Z) . f is V22() V23() ext-real Element of REAL
exp_R . f is V22() V23() ext-real Element of REAL
1 + (exp_R . f) is V22() V23() ext-real Element of REAL
1 - (exp_R . f) is V22() V23() ext-real Element of REAL
(1 + (exp_R . f)) / (1 - (exp_R . f)) is V22() V23() ext-real Element of REAL
((#Z 2) * (x - exp_R)) . f is V22() V23() ext-real Element of REAL
(x - exp_R) . f is V22() V23() ext-real Element of REAL
(#Z 2) . ((x - exp_R) . f) is V22() V23() ext-real Element of REAL
((x - exp_R) . f) #Z 2 is V22() V23() ext-real Element of REAL
x . f is V22() V23() ext-real Element of REAL
(x . f) - (exp_R . f) is V22() V23() ext-real Element of REAL
((x . f) - (exp_R . f)) #Z 2 is V22() V23() ext-real Element of REAL
(1 - (exp_R . f)) #Z 2 is V22() V23() ext-real Element of REAL
(exp_R / ((#Z 2) * (x - exp_R))) . f is V22() V23() ext-real Element of REAL
(((#Z 2) * (x - exp_R)) . f) " is V22() V23() ext-real Element of REAL
(exp_R . f) * ((((#Z 2) * (x - exp_R)) . f) ") is V22() V23() ext-real Element of REAL
1 / (((#Z 2) * (x - exp_R)) . f) is V22() V23() ext-real Element of REAL
(exp_R . f) * (1 / (((#Z 2) * (x - exp_R)) . f)) is V22() V23() ext-real Element of REAL
(exp_R . f) / ((1 - (exp_R . f)) #Z 2) is V22() V23() ext-real Element of REAL
diff ((exp_R / ((#Z 2) * (x - exp_R))),f) is V22() V23() ext-real Element of REAL
diff (exp_R,f) is V22() V23() ext-real Element of REAL
(diff (exp_R,f)) * (((#Z 2) * (x - exp_R)) . f) is V22() V23() ext-real Element of REAL
diff (((#Z 2) * (x - exp_R)),f) is V22() V23() ext-real Element of REAL
(diff (((#Z 2) * (x - exp_R)),f)) * (exp_R . f) is V22() V23() ext-real Element of REAL
((diff (exp_R,f)) * (((#Z 2) * (x - exp_R)) . f)) - ((diff (((#Z 2) * (x - exp_R)),f)) * (exp_R . f)) is V22() V23() ext-real Element of REAL
(((#Z 2) * (x - exp_R)) . f) ^2 is V22() V23() ext-real Element of REAL
K57((((#Z 2) * (x - exp_R)) . f),(((#Z 2) * (x - exp_R)) . f)) is set
(((diff (exp_R,f)) * (((#Z 2) * (x - exp_R)) . f)) - ((diff (((#Z 2) * (x - exp_R)),f)) * (exp_R . f))) / ((((#Z 2) * (x - exp_R)) . f) ^2) is V22() V23() ext-real Element of REAL
(exp_R . f) * (((#Z 2) * (x - exp_R)) . f) is V22() V23() ext-real Element of REAL
((exp_R . f) * (((#Z 2) * (x - exp_R)) . f)) - ((diff (((#Z 2) * (x - exp_R)),f)) * (exp_R . f)) is V22() V23() ext-real Element of REAL
(((exp_R . f) * (((#Z 2) * (x - exp_R)) . f)) - ((diff (((#Z 2) * (x - exp_R)),f)) * (exp_R . f))) / ((((#Z 2) * (x - exp_R)) . f) ^2) is V22() V23() ext-real Element of REAL
((#Z 2) * (x - exp_R)) `| Z is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
(((#Z 2) * (x - exp_R)) `| Z) . f is V22() V23() ext-real Element of REAL
((((#Z 2) * (x - exp_R)) `| Z) . f) * (exp_R . f) is V22() V23() ext-real Element of REAL
((exp_R . f) * (((#Z 2) * (x - exp_R)) . f)) - (((((#Z 2) * (x - exp_R)) `| Z) . f) * (exp_R . f)) is V22() V23() ext-real Element of REAL
(((exp_R . f) * (((#Z 2) * (x - exp_R)) . f)) - (((((#Z 2) * (x - exp_R)) `| Z) . f) * (exp_R . f))) / ((((#Z 2) * (x - exp_R)) . f) ^2) is V22() V23() ext-real Element of REAL
(exp_R . f) * ((1 - (exp_R . f)) #Z 2) is V22() V23() ext-real Element of REAL
2 * (exp_R . f) is V22() V23() ext-real Element of REAL
(2 * (exp_R . f)) * (1 - (exp_R . f)) is V22() V23() ext-real Element of REAL
- ((2 * (exp_R . f)) * (1 - (exp_R . f))) is V22() V23() ext-real Element of REAL
(- ((2 * (exp_R . f)) * (1 - (exp_R . f)))) * (exp_R . f) is V22() V23() ext-real Element of REAL
((exp_R . f) * ((1 - (exp_R . f)) #Z 2)) - ((- ((2 * (exp_R . f)) * (1 - (exp_R . f)))) * (exp_R . f)) is V22() V23() ext-real Element of REAL
((1 - (exp_R . f)) #Z 2) ^2 is V22() V23() ext-real Element of REAL
K57(((1 - (exp_R . f)) #Z 2),((1 - (exp_R . f)) #Z 2)) is set
(((exp_R . f) * ((1 - (exp_R . f)) #Z 2)) - ((- ((2 * (exp_R . f)) * (1 - (exp_R . f)))) * (exp_R . f))) / (((1 - (exp_R . f)) #Z 2) ^2) is V22() V23() ext-real Element of REAL
2 * (1 - (exp_R . f)) is V22() V23() ext-real Element of REAL
(2 * (1 - (exp_R . f))) * (exp_R . f) is V22() V23() ext-real Element of REAL
((1 - (exp_R . f)) #Z 2) + ((2 * (1 - (exp_R . f))) * (exp_R . f)) is V22() V23() ext-real Element of REAL
(exp_R . f) * (((1 - (exp_R . f)) #Z 2) + ((2 * (1 - (exp_R . f))) * (exp_R . f))) is V22() V23() ext-real Element of REAL
((1 - (exp_R . f)) #Z 2) * ((1 - (exp_R . f)) #Z 2) is V22() V23() ext-real Element of REAL
((exp_R . f) * (((1 - (exp_R . f)) #Z 2) + ((2 * (1 - (exp_R . f))) * (exp_R . f)))) / (((1 - (exp_R . f)) #Z 2) * ((1 - (exp_R . f)) #Z 2)) is V22() V23() ext-real Element of REAL
((exp_R . f) / ((1 - (exp_R . f)) #Z 2)) * (((1 - (exp_R . f)) #Z 2) + ((2 * (1 - (exp_R . f))) * (exp_R . f))) is V22() V23() ext-real Element of REAL
(((exp_R . f) / ((1 - (exp_R . f)) #Z 2)) * (((1 - (exp_R . f)) #Z 2) + ((2 * (1 - (exp_R . f))) * (exp_R . f)))) / ((1 - (exp_R . f)) #Z 2) is V22() V23() ext-real Element of REAL
1 + 1 is V15() V16() V17() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() Element of NAT
(1 - (exp_R . f)) #Z (1 + 1) is V22() V23() ext-real Element of REAL
(1 - (exp_R . f)) #Z 1 is V22() V23() ext-real Element of REAL
((1 - (exp_R . f)) #Z 1) * ((1 - (exp_R . f)) #Z 1) is V22() V23() ext-real Element of REAL
(1 - (exp_R . f)) * ((1 - (exp_R . f)) #Z 1) is V22() V23() ext-real Element of REAL
(1 - (exp_R . f)) * (1 - (exp_R . f)) is V22() V23() ext-real Element of REAL
diff ((ln * (exp_R / ((#Z 2) * (x - exp_R)))),f) is V22() V23() ext-real Element of REAL
((((exp_R . f) / ((1 - (exp_R . f)) #Z 2)) * (((1 - (exp_R . f)) #Z 2) + ((2 * (1 - (exp_R . f))) * (exp_R . f)))) / ((1 - (exp_R . f)) #Z 2)) / ((exp_R . f) / ((1 - (exp_R . f)) #Z 2)) is V22() V23() ext-real Element of REAL
((exp_R . f) / ((1 - (exp_R . f)) #Z 2)) * ((1 - (exp_R . f)) #Z 2) is V22() V23() ext-real Element of REAL
(((exp_R . f) / ((1 - (exp_R . f)) #Z 2)) * (((1 - (exp_R . f)) #Z 2) + ((2 * (1 - (exp_R . f))) * (exp_R . f)))) / (((exp_R . f) / ((1 - (exp_R . f)) #Z 2)) * ((1 - (exp_R . f)) #Z 2)) is V22() V23() ext-real Element of REAL
(1 - (exp_R . f)) * (1 + (exp_R . f)) is V22() V23() ext-real Element of REAL
((1 - (exp_R . f)) * (1 + (exp_R . f))) / ((1 - (exp_R . f)) * (1 - (exp_R . f))) is V22() V23() ext-real Element of REAL
f is V22() V23() ext-real Element of REAL
(x `| Z) . f is V22() V23() ext-real Element of REAL
exp_R . f is V22() V23() ext-real Element of REAL
1 + (exp_R . f) is V22() V23() ext-real Element of REAL
1 - (exp_R . f) is V22() V23() ext-real Element of REAL
(1 + (exp_R . f)) / (1 - (exp_R . f)) is V22() V23() ext-real Element of REAL
Z is open V49() V50() V51() Element of K19(REAL)
x is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
dom x is set
x `| Z is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
x is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
x + exp_R is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
(#Z 2) * (x + exp_R) is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
exp_R / ((#Z 2) * (x + exp_R)) is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
ln * (exp_R / ((#Z 2) * (x + exp_R))) is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
dom (exp_R / ((#Z 2) * (x + exp_R))) is set
f is set
dom ((#Z 2) * (x + exp_R)) is set
dom exp_R is set
((#Z 2) * (x + exp_R)) " {0} is set
(dom ((#Z 2) * (x + exp_R))) \ (((#Z 2) * (x + exp_R)) " {0}) is Element of K19((dom ((#Z 2) * (x + exp_R))))
K19((dom ((#Z 2) * (x + exp_R)))) is set
(dom exp_R) /\ ((dom ((#Z 2) * (x + exp_R))) \ (((#Z 2) * (x + exp_R)) " {0})) is Element of K19((dom ((#Z 2) * (x + exp_R))))
dom (x + exp_R) is set
f is set
f is V22() V23() ext-real Element of REAL
((#Z 2) * (x + exp_R)) . f is V22() V23() ext-real Element of REAL
(x + exp_R) . f is V22() V23() ext-real Element of REAL
x . f is V22() V23() ext-real Element of REAL
exp_R . f is V22() V23() ext-real Element of REAL
(x . f) + (exp_R . f) is V22() V23() ext-real Element of REAL
1 + (exp_R . f) is V22() V23() ext-real Element of REAL
(#Z 2) . ((x + exp_R) . f) is V22() V23() ext-real Element of REAL
((x + exp_R) . f) #Z 2 is V22() V23() ext-real Element of REAL
f is V22() V23() ext-real Element of REAL
(exp_R / ((#Z 2) * (x + exp_R))) . f is V22() V23() ext-real Element of REAL
exp_R . f is V22() V23() ext-real Element of REAL
((#Z 2) * (x + exp_R)) . f is V22() V23() ext-real Element of REAL
(((#Z 2) * (x + exp_R)) . f) " is V22() V23() ext-real Element of REAL
(exp_R . f) * ((((#Z 2) * (x + exp_R)) . f) ") is V22() V23() ext-real Element of REAL
1 / (((#Z 2) * (x + exp_R)) . f) is V22() V23() ext-real Element of REAL
(exp_R . f) * (1 / (((#Z 2) * (x + exp_R)) . f)) is V22() V23() ext-real Element of REAL
(exp_R . f) / (((#Z 2) * (x + exp_R)) . f) is V22() V23() ext-real Element of REAL
f is V22() V23() ext-real Element of REAL
((#Z 2) * (x + exp_R)) . f is V22() V23() ext-real Element of REAL
f is V22() V23() ext-real Element of REAL
(exp_R / ((#Z 2) * (x + exp_R))) . f is V22() V23() ext-real Element of REAL
f is V22() V23() ext-real Element of REAL
(x `| Z) . f is V22() V23() ext-real Element of REAL
exp_R . f is V22() V23() ext-real Element of REAL
1 - (exp_R . f) is V22() V23() ext-real Element of REAL
1 + (exp_R . f) is V22() V23() ext-real Element of REAL
(1 - (exp_R . f)) / (1 + (exp_R . f)) is V22() V23() ext-real Element of REAL
((#Z 2) * (x + exp_R)) . f is V22() V23() ext-real Element of REAL
(x + exp_R) . f is V22() V23() ext-real Element of REAL
(#Z 2) . ((x + exp_R) . f) is V22() V23() ext-real Element of REAL
((x + exp_R) . f) #Z 2 is V22() V23() ext-real Element of REAL
x . f is V22() V23() ext-real Element of REAL
(x . f) + (exp_R . f) is V22() V23() ext-real Element of REAL
((x . f) + (exp_R . f)) #Z 2 is V22() V23() ext-real Element of REAL
(1 + (exp_R . f)) #Z 2 is V22() V23() ext-real Element of REAL
diff ((exp_R / ((#Z 2) * (x + exp_R))),f) is V22() V23() ext-real Element of REAL
diff (exp_R,f) is V22() V23() ext-real Element of REAL
(diff (exp_R,f)) * (((#Z 2) * (x + exp_R)) . f) is V22() V23() ext-real Element of REAL
diff (((#Z 2) * (x + exp_R)),f) is V22() V23() ext-real Element of REAL
(diff (((#Z 2) * (x + exp_R)),f)) * (exp_R . f) is V22() V23() ext-real Element of REAL
((diff (exp_R,f)) * (((#Z 2) * (x + exp_R)) . f)) - ((diff (((#Z 2) * (x + exp_R)),f)) * (exp_R . f)) is V22() V23() ext-real Element of REAL
(((#Z 2) * (x + exp_R)) . f) ^2 is V22() V23() ext-real Element of REAL
K57((((#Z 2) * (x + exp_R)) . f),(((#Z 2) * (x + exp_R)) . f)) is set
(((diff (exp_R,f)) * (((#Z 2) * (x + exp_R)) . f)) - ((diff (((#Z 2) * (x + exp_R)),f)) * (exp_R . f))) / ((((#Z 2) * (x + exp_R)) . f) ^2) is V22() V23() ext-real Element of REAL
(exp_R . f) * (((#Z 2) * (x + exp_R)) . f) is V22() V23() ext-real Element of REAL
((exp_R . f) * (((#Z 2) * (x + exp_R)) . f)) - ((diff (((#Z 2) * (x + exp_R)),f)) * (exp_R . f)) is V22() V23() ext-real Element of REAL
(((exp_R . f) * (((#Z 2) * (x + exp_R)) . f)) - ((diff (((#Z 2) * (x + exp_R)),f)) * (exp_R . f))) / ((((#Z 2) * (x + exp_R)) . f) ^2) is V22() V23() ext-real Element of REAL
((#Z 2) * (x + exp_R)) `| Z is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
(((#Z 2) * (x + exp_R)) `| Z) . f is V22() V23() ext-real Element of REAL
((((#Z 2) * (x + exp_R)) `| Z) . f) * (exp_R . f) is V22() V23() ext-real Element of REAL
((exp_R . f) * (((#Z 2) * (x + exp_R)) . f)) - (((((#Z 2) * (x + exp_R)) `| Z) . f) * (exp_R . f)) is V22() V23() ext-real Element of REAL
(((exp_R . f) * (((#Z 2) * (x + exp_R)) . f)) - (((((#Z 2) * (x + exp_R)) `| Z) . f) * (exp_R . f))) / ((((#Z 2) * (x + exp_R)) . f) ^2) is V22() V23() ext-real Element of REAL
(exp_R . f) * ((1 + (exp_R . f)) #Z 2) is V22() V23() ext-real Element of REAL
2 * (exp_R . f) is V22() V23() ext-real Element of REAL
(2 * (exp_R . f)) * (1 + (exp_R . f)) is V22() V23() ext-real Element of REAL
((2 * (exp_R . f)) * (1 + (exp_R . f))) * (exp_R . f) is V22() V23() ext-real Element of REAL
((exp_R . f) * ((1 + (exp_R . f)) #Z 2)) - (((2 * (exp_R . f)) * (1 + (exp_R . f))) * (exp_R . f)) is V22() V23() ext-real Element of REAL
((1 + (exp_R . f)) #Z 2) ^2 is V22() V23() ext-real Element of REAL
K57(((1 + (exp_R . f)) #Z 2),((1 + (exp_R . f)) #Z 2)) is set
(((exp_R . f) * ((1 + (exp_R . f)) #Z 2)) - (((2 * (exp_R . f)) * (1 + (exp_R . f))) * (exp_R . f))) / (((1 + (exp_R . f)) #Z 2) ^2) is V22() V23() ext-real Element of REAL
2 * (1 + (exp_R . f)) is V22() V23() ext-real Element of REAL
(2 * (1 + (exp_R . f))) * (exp_R . f) is V22() V23() ext-real Element of REAL
((1 + (exp_R . f)) #Z 2) - ((2 * (1 + (exp_R . f))) * (exp_R . f)) is V22() V23() ext-real Element of REAL
(exp_R . f) * (((1 + (exp_R . f)) #Z 2) - ((2 * (1 + (exp_R . f))) * (exp_R . f))) is V22() V23() ext-real Element of REAL
((1 + (exp_R . f)) #Z 2) * ((1 + (exp_R . f)) #Z 2) is V22() V23() ext-real Element of REAL
((exp_R . f) * (((1 + (exp_R . f)) #Z 2) - ((2 * (1 + (exp_R . f))) * (exp_R . f)))) / (((1 + (exp_R . f)) #Z 2) * ((1 + (exp_R . f)) #Z 2)) is V22() V23() ext-real Element of REAL
(exp_R . f) / ((1 + (exp_R . f)) #Z 2) is V22() V23() ext-real Element of REAL
((exp_R . f) / ((1 + (exp_R . f)) #Z 2)) * (((1 + (exp_R . f)) #Z 2) - ((2 * (1 + (exp_R . f))) * (exp_R . f))) is V22() V23() ext-real Element of REAL
(((exp_R . f) / ((1 + (exp_R . f)) #Z 2)) * (((1 + (exp_R . f)) #Z 2) - ((2 * (1 + (exp_R . f))) * (exp_R . f)))) / ((1 + (exp_R . f)) #Z 2) is V22() V23() ext-real Element of REAL
(exp_R / ((#Z 2) * (x + exp_R))) . f is V22() V23() ext-real Element of REAL
(((#Z 2) * (x + exp_R)) . f) " is V22() V23() ext-real Element of REAL
(exp_R . f) * ((((#Z 2) * (x + exp_R)) . f) ") is V22() V23() ext-real Element of REAL
1 / (((#Z 2) * (x + exp_R)) . f) is V22() V23() ext-real Element of REAL
(exp_R . f) * (1 / (((#Z 2) * (x + exp_R)) . f)) is V22() V23() ext-real Element of REAL
1 + 1 is V15() V16() V17() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() Element of NAT
(1 + (exp_R . f)) #Z (1 + 1) is V22() V23() ext-real Element of REAL
(1 + (exp_R . f)) #Z 1 is V22() V23() ext-real Element of REAL
((1 + (exp_R . f)) #Z 1) * ((1 + (exp_R . f)) #Z 1) is V22() V23() ext-real Element of REAL
(1 + (exp_R . f)) * ((1 + (exp_R . f)) #Z 1) is V22() V23() ext-real Element of REAL
(1 + (exp_R . f)) * (1 + (exp_R . f)) is V22() V23() ext-real Element of REAL
diff ((ln * (exp_R / ((#Z 2) * (x + exp_R)))),f) is V22() V23() ext-real Element of REAL
((((exp_R . f) / ((1 + (exp_R . f)) #Z 2)) * (((1 + (exp_R . f)) #Z 2) - ((2 * (1 + (exp_R . f))) * (exp_R . f)))) / ((1 + (exp_R . f)) #Z 2)) / ((exp_R . f) / ((1 + (exp_R . f)) #Z 2)) is V22() V23() ext-real Element of REAL
((exp_R . f) / ((1 + (exp_R . f)) #Z 2)) * ((1 + (exp_R . f)) #Z 2) is V22() V23() ext-real Element of REAL
(((exp_R . f) / ((1 + (exp_R . f)) #Z 2)) * (((1 + (exp_R . f)) #Z 2) - ((2 * (1 + (exp_R . f))) * (exp_R . f)))) / (((exp_R . f) / ((1 + (exp_R . f)) #Z 2)) * ((1 + (exp_R . f)) #Z 2)) is V22() V23() ext-real Element of REAL
(1 + (exp_R . f)) * (1 - (exp_R . f)) is V22() V23() ext-real Element of REAL
((1 + (exp_R . f)) * (1 - (exp_R . f))) / ((1 + (exp_R . f)) * (1 + (exp_R . f))) is V22() V23() ext-real Element of REAL
f is V22() V23() ext-real Element of REAL
(x `| Z) . f is V22() V23() ext-real Element of REAL
exp_R . f is V22() V23() ext-real Element of REAL
1 - (exp_R . f) is V22() V23() ext-real Element of REAL
1 + (exp_R . f) is V22() V23() ext-real Element of REAL
(1 - (exp_R . f)) / (1 + (exp_R . f)) is V22() V23() ext-real Element of REAL
Z is open V49() V50() V51() Element of K19(REAL)
x is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
ln * x is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
dom (ln * x) is set
(ln * x) `| Z is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
x is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
exp_R * x is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
exp_R + (exp_R * x) is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
dom x is set
f is set
dom (exp_R + (exp_R * x)) is set
dom exp_R is set
dom (exp_R * x) is set
(dom exp_R) /\ (dom (exp_R * x)) is set
(exp_R + (exp_R * x)) `| Z is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
f is V22() V23() ext-real Element of REAL
((exp_R + (exp_R * x)) `| Z) . f is V22() V23() ext-real Element of REAL
exp_R f is V22() V23() ext-real Element of REAL
- f is V22() V23() ext-real Element of REAL
exp_R (- f) is V22() V23() ext-real Element of REAL
(exp_R f) - (exp_R (- f)) is V22() V23() ext-real Element of REAL
diff (exp_R,f) is V22() V23() ext-real Element of REAL
diff ((exp_R * x),f) is V22() V23() ext-real Element of REAL
(diff (exp_R,f)) + (diff ((exp_R * x),f)) is V22() V23() ext-real Element of REAL
exp_R . f is V22() V23() ext-real Element of REAL
(exp_R . f) + (diff ((exp_R * x),f)) is V22() V23() ext-real Element of REAL
(exp_R * x) `| Z is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
((exp_R * x) `| Z) . f is V22() V23() ext-real Element of REAL
(exp_R . f) + (((exp_R * x) `| Z) . f) is V22() V23() ext-real Element of REAL
- (exp_R (- f)) is V22() V23() ext-real Element of REAL
(exp_R . f) + (- (exp_R (- f))) is V22() V23() ext-real Element of REAL
(exp_R f) + (- (exp_R (- f))) is V22() V23() ext-real Element of REAL
f is V22() V23() ext-real Element of REAL
(exp_R + (exp_R * x)) . f is V22() V23() ext-real Element of REAL
exp_R f is V22() V23() ext-real Element of REAL
exp_R . f is V22() V23() ext-real Element of REAL
(exp_R * x) . f is V22() V23() ext-real Element of REAL
(exp_R . f) + ((exp_R * x) . f) is V22() V23() ext-real Element of REAL
x . f is V22() V23() ext-real Element of REAL
exp_R . (x . f) is V22() V23() ext-real Element of REAL
(exp_R . f) + (exp_R . (x . f)) is V22() V23() ext-real Element of REAL
- f is V22() V23() ext-real Element of REAL
exp_R . (- f) is V22() V23() ext-real Element of REAL
(exp_R . f) + (exp_R . (- f)) is V22() V23() ext-real Element of REAL
(exp_R f) + (exp_R . (- f)) is V22() V23() ext-real Element of REAL
exp_R (- f) is V22() V23() ext-real Element of REAL
(exp_R f) + (exp_R (- f)) is V22() V23() ext-real Element of REAL
f is V22() V23() ext-real Element of REAL
x . f is V22() V23() ext-real Element of REAL
f is V22() V23() ext-real Element of REAL
((ln * x) `| Z) . f is V22() V23() ext-real Element of REAL
exp_R f is V22() V23() ext-real Element of REAL
- f is V22() V23() ext-real Element of REAL
exp_R (- f) is V22() V23() ext-real Element of REAL
(exp_R f) - (exp_R (- f)) is V22() V23() ext-real Element of REAL
(exp_R f) + (exp_R (- f)) is V22() V23() ext-real Element of REAL
((exp_R f) - (exp_R (- f))) / ((exp_R f) + (exp_R (- f))) is V22() V23() ext-real Element of REAL
x . f is V22() V23() ext-real Element of REAL
exp_R . f is V22() V23() ext-real Element of REAL
(exp_R * x) . f is V22() V23() ext-real Element of REAL
(exp_R . f) + ((exp_R * x) . f) is V22() V23() ext-real Element of REAL
x . f is V22() V23() ext-real Element of REAL
exp_R . (x . f) is V22() V23() ext-real Element of REAL
(exp_R . f) + (exp_R . (x . f)) is V22() V23() ext-real Element of REAL
exp_R . (- f) is V22() V23() ext-real Element of REAL
(exp_R . f) + (exp_R . (- f)) is V22() V23() ext-real Element of REAL
(exp_R f) + (exp_R . (- f)) is V22() V23() ext-real Element of REAL
diff ((ln * x),f) is V22() V23() ext-real Element of REAL
diff (x,f) is V22() V23() ext-real Element of REAL
(diff (x,f)) / (x . f) is V22() V23() ext-real Element of REAL
x `| Z is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
(x `| Z) . f is V22() V23() ext-real Element of REAL
((x `| Z) . f) / (x . f) is V22() V23() ext-real Element of REAL
f is V22() V23() ext-real Element of REAL
((ln * x) `| Z) . f is V22() V23() ext-real Element of REAL
exp_R f is V22() V23() ext-real Element of REAL
- f is V22() V23() ext-real Element of REAL
exp_R (- f) is V22() V23() ext-real Element of REAL
(exp_R f) - (exp_R (- f)) is V22() V23() ext-real Element of REAL
(exp_R f) + (exp_R (- f)) is V22() V23() ext-real Element of REAL
((exp_R f) - (exp_R (- f))) / ((exp_R f) + (exp_R (- f))) is V22() V23() ext-real Element of REAL
Z is open V49() V50() V51() Element of K19(REAL)
x is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
ln * x is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
dom (ln * x) is set
(ln * x) `| Z is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
x is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
exp_R * x is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
exp_R - (exp_R * x) is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
- (exp_R * x) is Relation-like V6() V39() set
K58(1) is V22() V23() V68() set
K58(1) (#) (exp_R * x) is Relation-like V6() set
exp_R + (- (exp_R * x)) is Relation-like V6() set
dom x is set
f is set
dom (exp_R - (exp_R * x)) is set
dom exp_R is set
dom (exp_R * x) is set
(dom exp_R) /\ (dom (exp_R * x)) is set
f is V22() V23() ext-real Element of REAL
x . f is V22() V23() ext-real Element of REAL
- f is V22() V23() ext-real Element of REAL
(exp_R - (exp_R * x)) `| Z is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
f is V22() V23() ext-real Element of REAL
((exp_R - (exp_R * x)) `| Z) . f is V22() V23() ext-real Element of REAL
exp_R f is V22() V23() ext-real Element of REAL
- f is V22() V23() ext-real Element of REAL
exp_R (- f) is V22() V23() ext-real Element of REAL
(exp_R f) + (exp_R (- f)) is V22() V23() ext-real Element of REAL
diff (exp_R,f) is V22() V23() ext-real Element of REAL
diff ((exp_R * x),f) is V22() V23() ext-real Element of REAL
(diff (exp_R,f)) - (diff ((exp_R * x),f)) is V22() V23() ext-real Element of REAL
exp_R . f is V22() V23() ext-real Element of REAL
(exp_R . f) - (diff ((exp_R * x),f)) is V22() V23() ext-real Element of REAL
(exp_R * x) `| Z is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
((exp_R * x) `| Z) . f is V22() V23() ext-real Element of REAL
(exp_R . f) - (((exp_R * x) `| Z) . f) is V22() V23() ext-real Element of REAL
- (exp_R (- f)) is V22() V23() ext-real Element of REAL
(exp_R . f) - (- (exp_R (- f))) is V22() V23() ext-real Element of REAL
(exp_R . f) + (exp_R (- f)) is V22() V23() ext-real Element of REAL
f is V22() V23() ext-real Element of REAL
x . f is V22() V23() ext-real Element of REAL
f is V22() V23() ext-real Element of REAL
((ln * x) `| Z) . f is V22() V23() ext-real Element of REAL
exp_R f is V22() V23() ext-real Element of REAL
- f is V22() V23() ext-real Element of REAL
exp_R (- f) is V22() V23() ext-real Element of REAL
(exp_R f) + (exp_R (- f)) is V22() V23() ext-real Element of REAL
(exp_R f) - (exp_R (- f)) is V22() V23() ext-real Element of REAL
((exp_R f) + (exp_R (- f))) / ((exp_R f) - (exp_R (- f))) is V22() V23() ext-real Element of REAL
x . f is V22() V23() ext-real Element of REAL
exp_R . f is V22() V23() ext-real Element of REAL
(exp_R * x) . f is V22() V23() ext-real Element of REAL
(exp_R . f) - ((exp_R * x) . f) is V22() V23() ext-real Element of REAL
x . f is V22() V23() ext-real Element of REAL
exp_R . (x . f) is V22() V23() ext-real Element of REAL
(exp_R . f) - (exp_R . (x . f)) is V22() V23() ext-real Element of REAL
exp_R . (- f) is V22() V23() ext-real Element of REAL
(exp_R . f) - (exp_R . (- f)) is V22() V23() ext-real Element of REAL
(exp_R f) - (exp_R . (- f)) is V22() V23() ext-real Element of REAL
diff ((ln * x),f) is V22() V23() ext-real Element of REAL
diff (x,f) is V22() V23() ext-real Element of REAL
(diff (x,f)) / (x . f) is V22() V23() ext-real Element of REAL
x `| Z is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
(x `| Z) . f is V22() V23() ext-real Element of REAL
((x `| Z) . f) / (x . f) is V22() V23() ext-real Element of REAL
f is V22() V23() ext-real Element of REAL
((ln * x) `| Z) . f is V22() V23() ext-real Element of REAL
exp_R f is V22() V23() ext-real Element of REAL
- f is V22() V23() ext-real Element of REAL
exp_R (- f) is V22() V23() ext-real Element of REAL
(exp_R f) + (exp_R (- f)) is V22() V23() ext-real Element of REAL
(exp_R f) - (exp_R (- f)) is V22() V23() ext-real Element of REAL
((exp_R f) + (exp_R (- f))) / ((exp_R f) - (exp_R (- f))) is V22() V23() ext-real Element of REAL
3 is non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() Element of NAT
3 / 2 is V22() V23() ext-real Element of REAL
#R (3 / 2) is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
2 / 3 is V22() V23() ext-real Element of REAL
Z is open V49() V50() V51() Element of K19(REAL)
x is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
x + exp_R is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
(#R (3 / 2)) * (x + exp_R) is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
(2 / 3) (#) ((#R (3 / 2)) * (x + exp_R)) is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
dom ((2 / 3) (#) ((#R (3 / 2)) * (x + exp_R))) is set
((2 / 3) (#) ((#R (3 / 2)) * (x + exp_R))) `| Z is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
x is V22() V23() ext-real Element of REAL
x . x is V22() V23() ext-real Element of REAL
0 * x is V22() V23() ext-real Element of REAL
(0 * x) + 1 is V22() V23() ext-real Element of REAL
dom ((#R (3 / 2)) * (x + exp_R)) is set
dom (x + exp_R) is set
x is set
dom exp_R is set
dom x is set
(dom exp_R) /\ (dom x) is set
x is V22() V23() ext-real Element of REAL
(x + exp_R) . x is V22() V23() ext-real Element of REAL
x . x is V22() V23() ext-real Element of REAL
exp_R . x is V22() V23() ext-real Element of REAL
(x . x) + (exp_R . x) is V22() V23() ext-real Element of REAL
1 + (exp_R . x) is V22() V23() ext-real Element of REAL
x is V22() V23() ext-real Element of REAL
(x + exp_R) . x is V22() V23() ext-real Element of REAL
x is V22() V23() ext-real Element of REAL
(((2 / 3) (#) ((#R (3 / 2)) * (x + exp_R))) `| Z) . x is V22() V23() ext-real Element of REAL
exp_R . x is V22() V23() ext-real Element of REAL
1 + (exp_R . x) is V22() V23() ext-real Element of REAL
(1 + (exp_R . x)) #R (1 / 2) is V22() V23() ext-real Element of REAL
(exp_R . x) * ((1 + (exp_R . x)) #R (1 / 2)) is V22() V23() ext-real Element of REAL
(x + exp_R) `| Z is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
((x + exp_R) `| Z) . x is V22() V23() ext-real Element of REAL
diff (x,x) is V22() V23() ext-real Element of REAL
diff (exp_R,x) is V22() V23() ext-real Element of REAL
(diff (x,x)) + (diff (exp_R,x)) is V22() V23() ext-real Element of REAL
(diff (x,x)) + (exp_R . x) is V22() V23() ext-real Element of REAL
x `| Z is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
(x `| Z) . x is V22() V23() ext-real Element of REAL
((x `| Z) . x) + (exp_R . x) is V22() V23() ext-real Element of REAL
0 + (exp_R . x) is V22() V23() ext-real Element of REAL
(x + exp_R) . x is V22() V23() ext-real Element of REAL
x . x is V22() V23() ext-real Element of REAL
(x . x) + (exp_R . x) is V22() V23() ext-real Element of REAL
diff (((#R (3 / 2)) * (x + exp_R)),x) is V22() V23() ext-real Element of REAL
(2 / 3) * (diff (((#R (3 / 2)) * (x + exp_R)),x)) is V22() V23() ext-real Element of REAL
(3 / 2) - 1 is V22() V23() ext-real Element of REAL
((x + exp_R) . x) #R ((3 / 2) - 1) is V22() V23() ext-real Element of REAL
(3 / 2) * (((x + exp_R) . x) #R ((3 / 2) - 1)) is V22() V23() ext-real Element of REAL
diff ((x + exp_R),x) is V22() V23() ext-real Element of REAL
((3 / 2) * (((x + exp_R) . x) #R ((3 / 2) - 1))) * (diff ((x + exp_R),x)) is V22() V23() ext-real Element of REAL
(2 / 3) * (((3 / 2) * (((x + exp_R) . x) #R ((3 / 2) - 1))) * (diff ((x + exp_R),x))) is V22() V23() ext-real Element of REAL
((3 / 2) * (((x + exp_R) . x) #R ((3 / 2) - 1))) * (((x + exp_R) `| Z) . x) is V22() V23() ext-real Element of REAL
(2 / 3) * (((3 / 2) * (((x + exp_R) . x) #R ((3 / 2) - 1))) * (((x + exp_R) `| Z) . x)) is V22() V23() ext-real Element of REAL
x is V22() V23() ext-real Element of REAL
(((2 / 3) (#) ((#R (3 / 2)) * (x + exp_R))) `| Z) . x is V22() V23() ext-real Element of REAL
exp_R . x is V22() V23() ext-real Element of REAL
1 + (exp_R . x) is V22() V23() ext-real Element of REAL
(1 + (exp_R . x)) #R (1 / 2) is V22() V23() ext-real Element of REAL
(exp_R . x) * ((1 + (exp_R . x)) #R (1 / 2)) is V22() V23() ext-real Element of REAL
Z is V22() V23() ext-real Element of REAL
log (number_e,Z) is V22() V23() ext-real Element of REAL
3 * (log (number_e,Z)) is V22() V23() ext-real Element of REAL
2 / (3 * (log (number_e,Z))) is V22() V23() ext-real Element of REAL
x is open V49() V50() V51() Element of K19(REAL)
x is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
f is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
exp_R * f is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
x + (exp_R * f) is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
(#R (3 / 2)) * (x + (exp_R * f)) is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
(2 / (3 * (log (number_e,Z)))) (#) ((#R (3 / 2)) * (x + (exp_R * f))) is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
dom ((2 / (3 * (log (number_e,Z)))) (#) ((#R (3 / 2)) * (x + (exp_R * f)))) is set
((2 / (3 * (log (number_e,Z)))) (#) ((#R (3 / 2)) * (x + (exp_R * f)))) `| x is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
x is V22() V23() ext-real Element of REAL
x . x is V22() V23() ext-real Element of REAL
0 * x is V22() V23() ext-real Element of REAL
(0 * x) + 1 is V22() V23() ext-real Element of REAL
dom ((#R (3 / 2)) * (x + (exp_R * f))) is set
dom (x + (exp_R * f)) is set
x is set
dom (exp_R * f) is set
dom x is set
(dom (exp_R * f)) /\ (dom x) is set
x is V22() V23() ext-real Element of REAL
f . x is V22() V23() ext-real Element of REAL
x * (log (number_e,Z)) is V22() V23() ext-real Element of REAL
x is V22() V23() ext-real Element of REAL
(x + (exp_R * f)) . x is V22() V23() ext-real Element of REAL
x . x is V22() V23() ext-real Element of REAL
(exp_R * f) . x is V22() V23() ext-real Element of REAL
(x . x) + ((exp_R * f) . x) is V22() V23() ext-real Element of REAL
f . x is V22() V23() ext-real Element of REAL
exp_R . (f . x) is V22() V23() ext-real Element of REAL
(x . x) + (exp_R . (f . x)) is V22() V23() ext-real Element of REAL
1 + (exp_R . (f . x)) is V22() V23() ext-real Element of REAL
x * (log (number_e,Z)) is V22() V23() ext-real Element of REAL
exp_R . (x * (log (number_e,Z))) is V22() V23() ext-real Element of REAL
1 + (exp_R . (x * (log (number_e,Z)))) is V22() V23() ext-real Element of REAL
x is V22() V23() ext-real Element of REAL
(x + (exp_R * f)) . x is V22() V23() ext-real Element of REAL
log (number_e,1) is V22() V23() ext-real Element of REAL
number_e to_power (log (number_e,1)) is V22() V23() ext-real Element of REAL
x is V22() V23() ext-real Element of REAL
(((2 / (3 * (log (number_e,Z)))) (#) ((#R (3 / 2)) * (x + (exp_R * f)))) `| x) . x is V22() V23() ext-real Element of REAL
Z #R x is V22() V23() ext-real Element of REAL
1 + (Z #R x) is V22() V23() ext-real Element of REAL
(1 + (Z #R x)) #R (1 / 2) is V22() V23() ext-real Element of REAL
(Z #R x) * ((1 + (Z #R x)) #R (1 / 2)) is V22() V23() ext-real Element of REAL
(x + (exp_R * f)) `| x is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
((x + (exp_R * f)) `| x) . x is V22() V23() ext-real Element of REAL
diff (x,x) is V22() V23() ext-real Element of REAL
diff ((exp_R * f),x) is V22() V23() ext-real Element of REAL
(diff (x,x)) + (diff ((exp_R * f),x)) is V22() V23() ext-real Element of REAL
(exp_R * f) `| x is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
((exp_R * f) `| x) . x is V22() V23() ext-real Element of REAL
(diff (x,x)) + (((exp_R * f) `| x) . x) is V22() V23() ext-real Element of REAL
x `| x is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
(x `| x) . x is V22() V23() ext-real Element of REAL
((x `| x) . x) + (((exp_R * f) `| x) . x) is V22() V23() ext-real Element of REAL
0 + (((exp_R * f) `| x) . x) is V22() V23() ext-real Element of REAL
(Z #R x) * (log (number_e,Z)) is V22() V23() ext-real Element of REAL
(x + (exp_R * f)) . x is V22() V23() ext-real Element of REAL
x . x is V22() V23() ext-real Element of REAL
(exp_R * f) . x is V22() V23() ext-real Element of REAL
(x . x) + ((exp_R * f) . x) is V22() V23() ext-real Element of REAL
f . x is V22() V23() ext-real Element of REAL
exp_R . (f . x) is V22() V23() ext-real Element of REAL
(x . x) + (exp_R . (f . x)) is V22() V23() ext-real Element of REAL
1 + (exp_R . (f . x)) is V22() V23() ext-real Element of REAL
x * (log (number_e,Z)) is V22() V23() ext-real Element of REAL
exp_R . (x * (log (number_e,Z))) is V22() V23() ext-real Element of REAL
1 + (exp_R . (x * (log (number_e,Z)))) is V22() V23() ext-real Element of REAL
diff (((#R (3 / 2)) * (x + (exp_R * f))),x) is V22() V23() ext-real Element of REAL
(3 / 2) - 1 is V22() V23() ext-real Element of REAL
((x + (exp_R * f)) . x) #R ((3 / 2) - 1) is V22() V23() ext-real Element of REAL
(3 / 2) * (((x + (exp_R * f)) . x) #R ((3 / 2) - 1)) is V22() V23() ext-real Element of REAL
diff ((x + (exp_R * f)),x) is V22() V23() ext-real Element of REAL
((3 / 2) * (((x + (exp_R * f)) . x) #R ((3 / 2) - 1))) * (diff ((x + (exp_R * f)),x)) is V22() V23() ext-real Element of REAL
(3 / 2) * ((1 + (Z #R x)) #R (1 / 2)) is V22() V23() ext-real Element of REAL
((3 / 2) * ((1 + (Z #R x)) #R (1 / 2))) * ((Z #R x) * (log (number_e,Z))) is V22() V23() ext-real Element of REAL
(3 * (log (number_e,Z))) / 2 is V22() V23() ext-real Element of REAL
((3 * (log (number_e,Z))) / 2) * (Z #R x) is V22() V23() ext-real Element of REAL
(((3 * (log (number_e,Z))) / 2) * (Z #R x)) * ((1 + (Z #R x)) #R (1 / 2)) is V22() V23() ext-real Element of REAL
(2 / (3 * (log (number_e,Z)))) * ((((3 * (log (number_e,Z))) / 2) * (Z #R x)) * ((1 + (Z #R x)) #R (1 / 2))) is V22() V23() ext-real Element of REAL
(2 / (3 * (log (number_e,Z)))) * ((3 * (log (number_e,Z))) / 2) is V22() V23() ext-real Element of REAL
((2 / (3 * (log (number_e,Z)))) * ((3 * (log (number_e,Z))) / 2)) * (Z #R x) is V22() V23() ext-real Element of REAL
(((2 / (3 * (log (number_e,Z)))) * ((3 * (log (number_e,Z))) / 2)) * (Z #R x)) * ((1 + (Z #R x)) #R (1 / 2)) is V22() V23() ext-real Element of REAL
1 * (Z #R x) is V22() V23() ext-real Element of REAL
(1 * (Z #R x)) * ((1 + (Z #R x)) #R (1 / 2)) is V22() V23() ext-real Element of REAL
x is V22() V23() ext-real Element of REAL
(((2 / (3 * (log (number_e,Z)))) (#) ((#R (3 / 2)) * (x + (exp_R * f)))) `| x) . x is V22() V23() ext-real Element of REAL
Z #R x is V22() V23() ext-real Element of REAL
1 + (Z #R x) is V22() V23() ext-real Element of REAL
(1 + (Z #R x)) #R (1 / 2) is V22() V23() ext-real Element of REAL
(Z #R x) * ((1 + (Z #R x)) #R (1 / 2)) is V22() V23() ext-real Element of REAL
- (1 / 2) is V22() V23() ext-real Element of REAL
Z is open V49() V50() V51() Element of K19(REAL)
x is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
cos * x is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
(- (1 / 2)) (#) (cos * x) is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
dom ((- (1 / 2)) (#) (cos * x)) is set
((- (1 / 2)) (#) (cos * x)) `| Z is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
dom (cos * x) is set
x is V22() V23() ext-real Element of REAL
x . x is V22() V23() ext-real Element of REAL
2 * x is V22() V23() ext-real Element of REAL
(2 * x) + 0 is V22() V23() ext-real Element of REAL
x is V22() V23() ext-real Element of REAL
(((- (1 / 2)) (#) (cos * x)) `| Z) . x is V22() V23() ext-real Element of REAL
2 * x is V22() V23() ext-real Element of REAL
sin (2 * x) is V22() V23() ext-real Element of REAL
diff ((cos * x),x) is V22() V23() ext-real Element of REAL
(- (1 / 2)) * (diff ((cos * x),x)) is V22() V23() ext-real Element of REAL
(cos * x) `| Z is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
((cos * x) `| Z) . x is V22() V23() ext-real Element of REAL
(- (1 / 2)) * (((cos * x) `| Z) . x) is V22() V23() ext-real Element of REAL
(2 * x) + 0 is V22() V23() ext-real Element of REAL
sin . ((2 * x) + 0) is V22() V23() ext-real Element of REAL
2 * (sin . ((2 * x) + 0)) is V22() V23() ext-real Element of REAL
- (2 * (sin . ((2 * x) + 0))) is V22() V23() ext-real Element of REAL
(- (1 / 2)) * (- (2 * (sin . ((2 * x) + 0)))) is V22() V23() ext-real Element of REAL
x is V22() V23() ext-real Element of REAL
(((- (1 / 2)) (#) (cos * x)) `| Z) . x is V22() V23() ext-real Element of REAL
2 * x is V22() V23() ext-real Element of REAL
sin (2 * x) is V22() V23() ext-real Element of REAL
#R (1 / 2) is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
Z is open V49() V50() V51() Element of K19(REAL)
x is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
x - cos is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
- cos is Relation-like V6() V39() set
K58(1) is V22() V23() V68() set
K58(1) (#) cos is Relation-like V6() set
x + (- cos) is Relation-like V6() set
(#R (1 / 2)) * (x - cos) is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
2 (#) ((#R (1 / 2)) * (x - cos)) is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
dom (2 (#) ((#R (1 / 2)) * (x - cos))) is set
(2 (#) ((#R (1 / 2)) * (x - cos))) `| Z is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
x is V22() V23() ext-real Element of REAL
x . x is V22() V23() ext-real Element of REAL
0 * x is V22() V23() ext-real Element of REAL
(0 * x) + 1 is V22() V23() ext-real Element of REAL
dom ((#R (1 / 2)) * (x - cos)) is set
dom (x - cos) is set
x is set
dom cos is set
dom x is set
(dom cos) /\ (dom x) is set
x is V22() V23() ext-real Element of REAL
(x - cos) . x is V22() V23() ext-real Element of REAL
cos . x is V22() V23() ext-real Element of REAL
1 - (cos . x) is V22() V23() ext-real Element of REAL
1 - 1 is V22() V23() ext-real V68() Element of REAL
x . x is V22() V23() ext-real Element of REAL
(x . x) - (cos . x) is V22() V23() ext-real Element of REAL
x is V22() V23() ext-real Element of REAL
(x - cos) . x is V22() V23() ext-real Element of REAL
x is V22() V23() ext-real Element of REAL
((2 (#) ((#R (1 / 2)) * (x - cos))) `| Z) . x is V22() V23() ext-real Element of REAL
cos . x is V22() V23() ext-real Element of REAL
1 + (cos . x) is V22() V23() ext-real Element of REAL
(1 + (cos . x)) #R (1 / 2) is V22() V23() ext-real Element of REAL
diff ((x - cos),x) is V22() V23() ext-real Element of REAL
(x - cos) `| Z is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
((x - cos) `| Z) . x is V22() V23() ext-real Element of REAL
diff (x,x) is V22() V23() ext-real Element of REAL
diff (cos,x) is V22() V23() ext-real Element of REAL
(diff (x,x)) - (diff (cos,x)) is V22() V23() ext-real Element of REAL
x `| Z is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
(x `| Z) . x is V22() V23() ext-real Element of REAL
((x `| Z) . x) - (diff (cos,x)) is V22() V23() ext-real Element of REAL
0 - (diff (cos,x)) is V22() V23() ext-real Element of REAL
sin . x is V22() V23() ext-real Element of REAL
- (sin . x) is V22() V23() ext-real Element of REAL
0 - (- (sin . x)) is V22() V23() ext-real Element of REAL
(x - cos) . x is V22() V23() ext-real Element of REAL
x . x is V22() V23() ext-real Element of REAL
(x . x) - (cos . x) is V22() V23() ext-real Element of REAL
1 - (cos . x) is V22() V23() ext-real Element of REAL
diff (((#R (1 / 2)) * (x - cos)),x) is V22() V23() ext-real Element of REAL
2 * (diff (((#R (1 / 2)) * (x - cos)),x)) is V22() V23() ext-real Element of REAL
(1 / 2) - 1 is V22() V23() ext-real Element of REAL
((x - cos) . x) #R ((1 / 2) - 1) is V22() V23() ext-real Element of REAL
(1 / 2) * (((x - cos) . x) #R ((1 / 2) - 1)) is V22() V23() ext-real Element of REAL
((1 / 2) * (((x - cos) . x) #R ((1 / 2) - 1))) * (diff ((x - cos),x)) is V22() V23() ext-real Element of REAL
2 * (((1 / 2) * (((x - cos) . x) #R ((1 / 2) - 1))) * (diff ((x - cos),x))) is V22() V23() ext-real Element of REAL
(1 - (cos . x)) #R (- (1 / 2)) is V22() V23() ext-real Element of REAL
(sin . x) * ((1 - (cos . x)) #R (- (1 / 2))) is V22() V23() ext-real Element of REAL
(1 - (cos . x)) #R (1 / 2) is V22() V23() ext-real Element of REAL
1 / ((1 - (cos . x)) #R (1 / 2)) is V22() V23() ext-real Element of REAL
(sin . x) * (1 / ((1 - (cos . x)) #R (1 / 2))) is V22() V23() ext-real Element of REAL
(sin . x) / ((1 - (cos . x)) #R (1 / 2)) is V22() V23() ext-real Element of REAL
x is V22() V23() ext-real Element of REAL
((2 (#) ((#R (1 / 2)) * (x - cos))) `| Z) . x is V22() V23() ext-real Element of REAL
cos . x is V22() V23() ext-real Element of REAL
1 + (cos . x) is V22() V23() ext-real Element of REAL
(1 + (cos . x)) #R (1 / 2) is V22() V23() ext-real Element of REAL
- 2 is V22() V23() ext-real V68() Element of REAL
Z is open V49() V50() V51() Element of K19(REAL)
x is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
x + cos is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
(#R (1 / 2)) * (x + cos) is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
(- 2) (#) ((#R (1 / 2)) * (x + cos)) is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
dom ((- 2) (#) ((#R (1 / 2)) * (x + cos))) is set
((- 2) (#) ((#R (1 / 2)) * (x + cos))) `| Z is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
x is V22() V23() ext-real Element of REAL
x . x is V22() V23() ext-real Element of REAL
0 * x is V22() V23() ext-real Element of REAL
(0 * x) + 1 is V22() V23() ext-real Element of REAL
dom ((#R (1 / 2)) * (x + cos)) is set
dom (x + cos) is set
x is set
dom cos is set
dom x is set
(dom cos) /\ (dom x) is set
x is V22() V23() ext-real Element of REAL
(x + cos) . x is V22() V23() ext-real Element of REAL
cos . x is V22() V23() ext-real Element of REAL
1 + (cos . x) is V22() V23() ext-real Element of REAL
1 + (- 1) is V22() V23() ext-real V68() Element of REAL
x . x is V22() V23() ext-real Element of REAL
(x . x) + (cos . x) is V22() V23() ext-real Element of REAL
x is V22() V23() ext-real Element of REAL
(x + cos) . x is V22() V23() ext-real Element of REAL
x is V22() V23() ext-real Element of REAL
(((- 2) (#) ((#R (1 / 2)) * (x + cos))) `| Z) . x is V22() V23() ext-real Element of REAL
cos . x is V22() V23() ext-real Element of REAL
1 - (cos . x) is V22() V23() ext-real Element of REAL
(1 - (cos . x)) #R (1 / 2) is V22() V23() ext-real Element of REAL
diff ((x + cos),x) is V22() V23() ext-real Element of REAL
(x + cos) `| Z is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
((x + cos) `| Z) . x is V22() V23() ext-real Element of REAL
diff (x,x) is V22() V23() ext-real Element of REAL
diff (cos,x) is V22() V23() ext-real Element of REAL
(diff (x,x)) + (diff (cos,x)) is V22() V23() ext-real Element of REAL
x `| Z is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
(x `| Z) . x is V22() V23() ext-real Element of REAL
((x `| Z) . x) + (diff (cos,x)) is V22() V23() ext-real Element of REAL
0 + (diff (cos,x)) is V22() V23() ext-real Element of REAL
sin . x is V22() V23() ext-real Element of REAL
- (sin . x) is V22() V23() ext-real Element of REAL
(x + cos) . x is V22() V23() ext-real Element of REAL
x . x is V22() V23() ext-real Element of REAL
(x . x) + (cos . x) is V22() V23() ext-real Element of REAL
1 + (cos . x) is V22() V23() ext-real Element of REAL
diff (((#R (1 / 2)) * (x + cos)),x) is V22() V23() ext-real Element of REAL
(- 2) * (diff (((#R (1 / 2)) * (x + cos)),x)) is V22() V23() ext-real Element of REAL
(1 / 2) - 1 is V22() V23() ext-real Element of REAL
((x + cos) . x) #R ((1 / 2) - 1) is V22() V23() ext-real Element of REAL
(1 / 2) * (((x + cos) . x) #R ((1 / 2) - 1)) is V22() V23() ext-real Element of REAL
((1 / 2) * (((x + cos) . x) #R ((1 / 2) - 1))) * (diff ((x + cos),x)) is V22() V23() ext-real Element of REAL
(- 2) * (((1 / 2) * (((x + cos) . x) #R ((1 / 2) - 1))) * (diff ((x + cos),x))) is V22() V23() ext-real Element of REAL
(1 + (cos . x)) #R (- (1 / 2)) is V22() V23() ext-real Element of REAL
(sin . x) * ((1 + (cos . x)) #R (- (1 / 2))) is V22() V23() ext-real Element of REAL
- ((sin . x) * ((1 + (cos . x)) #R (- (1 / 2)))) is V22() V23() ext-real Element of REAL
- (- ((sin . x) * ((1 + (cos . x)) #R (- (1 / 2))))) is V22() V23() ext-real Element of REAL
(1 + (cos . x)) #R (1 / 2) is V22() V23() ext-real Element of REAL
1 / ((1 + (cos . x)) #R (1 / 2)) is V22() V23() ext-real Element of REAL
(sin . x) * (1 / ((1 + (cos . x)) #R (1 / 2))) is V22() V23() ext-real Element of REAL
(sin . x) / ((1 + (cos . x)) #R (1 / 2)) is V22() V23() ext-real Element of REAL
x is V22() V23() ext-real Element of REAL
(((- 2) (#) ((#R (1 / 2)) * (x + cos))) `| Z) . x is V22() V23() ext-real Element of REAL
cos . x is V22() V23() ext-real Element of REAL
1 - (cos . x) is V22() V23() ext-real Element of REAL
(1 - (cos . x)) #R (1 / 2) is V22() V23() ext-real Element of REAL
2 (#) sin is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
Z is open V49() V50() V51() Element of K19(REAL)
x is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
x + (2 (#) sin) is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
dom (x + (2 (#) sin)) is set
(x + (2 (#) sin)) `| Z is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
dom x is set
dom (2 (#) sin) is set
(dom x) /\ (dom (2 (#) sin)) is set
x is V22() V23() ext-real Element of REAL
x . x is V22() V23() ext-real Element of REAL
0 * x is V22() V23() ext-real Element of REAL
(0 * x) + 1 is V22() V23() ext-real Element of REAL
x is V22() V23() ext-real Element of REAL
((x + (2 (#) sin)) `| Z) . x is V22() V23() ext-real Element of REAL
cos . x is V22() V23() ext-real Element of REAL
2 * (cos . x) is V22() V23() ext-real Element of REAL
diff (x,x) is V22() V23() ext-real Element of REAL
diff ((2 (#) sin),x) is V22() V23() ext-real Element of REAL
(diff (x,x)) + (diff ((2 (#) sin),x)) is V22() V23() ext-real Element of REAL
x `| Z is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
(x `| Z) . x is V22() V23() ext-real Element of REAL
((x `| Z) . x) + (diff ((2 (#) sin),x)) is V22() V23() ext-real Element of REAL
diff (sin,x) is V22() V23() ext-real Element of REAL
2 * (diff (sin,x)) is V22() V23() ext-real Element of REAL
((x `| Z) . x) + (2 * (diff (sin,x))) is V22() V23() ext-real Element of REAL
0 + (2 * (diff (sin,x))) is V22() V23() ext-real Element of REAL
x is V22() V23() ext-real Element of REAL
((x + (2 (#) sin)) `| Z) . x is V22() V23() ext-real Element of REAL
cos . x is V22() V23() ext-real Element of REAL
2 * (cos . x) is V22() V23() ext-real Element of REAL
Z is open V49() V50() V51() Element of K19(REAL)
x is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
ln * x is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
(1 / 2) (#) (ln * x) is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
dom ((1 / 2) (#) (ln * x)) is set
((1 / 2) (#) (ln * x)) `| Z is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
x is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
x + (2 (#) sin) is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
dom (ln * x) is set
dom x is set
f is set
dom (x + (2 (#) sin)) is set
f is V22() V23() ext-real Element of REAL
x . f is V22() V23() ext-real Element of REAL
f is V22() V23() ext-real Element of REAL
x . f is V22() V23() ext-real Element of REAL
dom x is set
dom (2 (#) sin) is set
(dom x) /\ (dom (2 (#) sin)) is set
f is V22() V23() ext-real Element of REAL
(((1 / 2) (#) (ln * x)) `| Z) . f is V22() V23() ext-real Element of REAL
cos . f is V22() V23() ext-real Element of REAL
sin . f is V22() V23() ext-real Element of REAL
2 * (sin . f) is V22() V23() ext-real Element of REAL
1 + (2 * (sin . f)) is V22() V23() ext-real Element of REAL
(cos . f) / (1 + (2 * (sin . f))) is V22() V23() ext-real Element of REAL
x . f is V22() V23() ext-real Element of REAL
x . f is V22() V23() ext-real Element of REAL
(2 (#) sin) . f is V22() V23() ext-real Element of REAL
(x . f) + ((2 (#) sin) . f) is V22() V23() ext-real Element of REAL
1 + ((2 (#) sin) . f) is V22() V23() ext-real Element of REAL
diff ((ln * x),f) is V22() V23() ext-real Element of REAL
(1 / 2) * (diff ((ln * x),f)) is V22() V23() ext-real Element of REAL
diff (x,f) is V22() V23() ext-real Element of REAL
(diff (x,f)) / (x . f) is V22() V23() ext-real Element of REAL
(1 / 2) * ((diff (x,f)) / (x . f)) is V22() V23() ext-real Element of REAL
x `| Z is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
(x `| Z) . f is V22() V23() ext-real Element of REAL
((x `| Z) . f) / (x . f) is V22() V23() ext-real Element of REAL
(1 / 2) * (((x `| Z) . f) / (x . f)) is V22() V23() ext-real Element of REAL
2 * (cos . f) is V22() V23() ext-real Element of REAL
(2 * (cos . f)) / (x . f) is V22() V23() ext-real Element of REAL
(1 / 2) * ((2 * (cos . f)) / (x . f)) is V22() V23() ext-real Element of REAL
(1 / 2) * (2 * (cos . f)) is V22() V23() ext-real Element of REAL
((1 / 2) * (2 * (cos . f))) / (x . f) is V22() V23() ext-real Element of REAL
f is V22() V23() ext-real Element of REAL
(((1 / 2) (#) (ln * x)) `| Z) . f is V22() V23() ext-real Element of REAL
cos . f is V22() V23() ext-real Element of REAL
sin . f is V22() V23() ext-real Element of REAL
2 * (sin . f) is V22() V23() ext-real Element of REAL
1 + (2 * (sin . f)) is V22() V23() ext-real Element of REAL
(cos . f) / (1 + (2 * (sin . f))) is V22() V23() ext-real Element of REAL
2 (#) cos is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
Z is open V49() V50() V51() Element of K19(REAL)
x is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
x + (2 (#) cos) is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
dom (x + (2 (#) cos)) is set
(x + (2 (#) cos)) `| Z is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
dom x is set
dom (2 (#) cos) is set
(dom x) /\ (dom (2 (#) cos)) is set
x is V22() V23() ext-real Element of REAL
x . x is V22() V23() ext-real Element of REAL
0 * x is V22() V23() ext-real Element of REAL
(0 * x) + 1 is V22() V23() ext-real Element of REAL
x is V22() V23() ext-real Element of REAL
((x + (2 (#) cos)) `| Z) . x is V22() V23() ext-real Element of REAL
sin . x is V22() V23() ext-real Element of REAL
2 * (sin . x) is V22() V23() ext-real Element of REAL
- (2 * (sin . x)) is V22() V23() ext-real Element of REAL
diff (x,x) is V22() V23() ext-real Element of REAL
diff ((2 (#) cos),x) is V22() V23() ext-real Element of REAL
(diff (x,x)) + (diff ((2 (#) cos),x)) is V22() V23() ext-real Element of REAL
x `| Z is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
(x `| Z) . x is V22() V23() ext-real Element of REAL
((x `| Z) . x) + (diff ((2 (#) cos),x)) is V22() V23() ext-real Element of REAL
diff (cos,x) is V22() V23() ext-real Element of REAL
2 * (diff (cos,x)) is V22() V23() ext-real Element of REAL
((x `| Z) . x) + (2 * (diff (cos,x))) is V22() V23() ext-real Element of REAL
0 + (2 * (diff (cos,x))) is V22() V23() ext-real Element of REAL
- (sin . x) is V22() V23() ext-real Element of REAL
2 * (- (sin . x)) is V22() V23() ext-real Element of REAL
0 + (2 * (- (sin . x))) is V22() V23() ext-real Element of REAL
x is V22() V23() ext-real Element of REAL
((x + (2 (#) cos)) `| Z) . x is V22() V23() ext-real Element of REAL
sin . x is V22() V23() ext-real Element of REAL
2 * (sin . x) is V22() V23() ext-real Element of REAL
- (2 * (sin . x)) is V22() V23() ext-real Element of REAL
Z is open V49() V50() V51() Element of K19(REAL)
x is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
ln * x is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
(- (1 / 2)) (#) (ln * x) is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
dom ((- (1 / 2)) (#) (ln * x)) is set
((- (1 / 2)) (#) (ln * x)) `| Z is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
x is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
x + (2 (#) cos) is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
dom (ln * x) is set
dom x is set
f is set
dom (x + (2 (#) cos)) is set
f is V22() V23() ext-real Element of REAL
x . f is V22() V23() ext-real Element of REAL
f is V22() V23() ext-real Element of REAL
x . f is V22() V23() ext-real Element of REAL
dom x is set
dom (2 (#) cos) is set
(dom x) /\ (dom (2 (#) cos)) is set
f is V22() V23() ext-real Element of REAL
(((- (1 / 2)) (#) (ln * x)) `| Z) . f is V22() V23() ext-real Element of REAL
sin . f is V22() V23() ext-real Element of REAL
cos . f is V22() V23() ext-real Element of REAL
2 * (cos . f) is V22() V23() ext-real Element of REAL
1 + (2 * (cos . f)) is V22() V23() ext-real Element of REAL
(sin . f) / (1 + (2 * (cos . f))) is V22() V23() ext-real Element of REAL
x . f is V22() V23() ext-real Element of REAL
x . f is V22() V23() ext-real Element of REAL
(2 (#) cos) . f is V22() V23() ext-real Element of REAL
(x . f) + ((2 (#) cos) . f) is V22() V23() ext-real Element of REAL
1 + ((2 (#) cos) . f) is V22() V23() ext-real Element of REAL
diff ((ln * x),f) is V22() V23() ext-real Element of REAL
(- (1 / 2)) * (diff ((ln * x),f)) is V22() V23() ext-real Element of REAL
diff (x,f) is V22() V23() ext-real Element of REAL
(diff (x,f)) / (x . f) is V22() V23() ext-real Element of REAL
(- (1 / 2)) * ((diff (x,f)) / (x . f)) is V22() V23() ext-real Element of REAL
x `| Z is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
(x `| Z) . f is V22() V23() ext-real Element of REAL
((x `| Z) . f) / (x . f) is V22() V23() ext-real Element of REAL
(- (1 / 2)) * (((x `| Z) . f) / (x . f)) is V22() V23() ext-real Element of REAL
2 * (sin . f) is V22() V23() ext-real Element of REAL
- (2 * (sin . f)) is V22() V23() ext-real Element of REAL
(- (2 * (sin . f))) / (x . f) is V22() V23() ext-real Element of REAL
(- (1 / 2)) * ((- (2 * (sin . f))) / (x . f)) is V22() V23() ext-real Element of REAL
(- (1 / 2)) * (- (2 * (sin . f))) is V22() V23() ext-real Element of REAL
((- (1 / 2)) * (- (2 * (sin . f)))) / (x . f) is V22() V23() ext-real Element of REAL
f is V22() V23() ext-real Element of REAL
(((- (1 / 2)) (#) (ln * x)) `| Z) . f is V22() V23() ext-real Element of REAL
sin . f is V22() V23() ext-real Element of REAL
cos . f is V22() V23() ext-real Element of REAL
2 * (cos . f) is V22() V23() ext-real Element of REAL
1 + (2 * (cos . f)) is V22() V23() ext-real Element of REAL
(sin . f) / (1 + (2 * (cos . f))) is V22() V23() ext-real Element of REAL
Z is V22() V23() ext-real Element of REAL
4 * Z is V22() V23() ext-real Element of REAL
1 / (4 * Z) is V22() V23() ext-real Element of REAL
2 * Z is V22() V23() ext-real Element of REAL
x is open V49() V50() V51() Element of K19(REAL)
x is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
sin * x is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
(1 / (4 * Z)) (#) (sin * x) is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
dom ((1 / (4 * Z)) (#) (sin * x)) is set
((1 / (4 * Z)) (#) (sin * x)) `| x is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
dom (sin * x) is set
f is V22() V23() ext-real Element of REAL
x . f is V22() V23() ext-real Element of REAL
(2 * Z) * f is V22() V23() ext-real Element of REAL
((2 * Z) * f) + 0 is V22() V23() ext-real Element of REAL
f is V22() V23() ext-real Element of REAL
(((1 / (4 * Z)) (#) (sin * x)) `| x) . f is V22() V23() ext-real Element of REAL
(2 * Z) * f is V22() V23() ext-real Element of REAL
cos ((2 * Z) * f) is V22() V23() ext-real Element of REAL
(1 / 2) * (cos ((2 * Z) * f)) is V22() V23() ext-real Element of REAL
diff ((sin * x),f) is V22() V23() ext-real Element of REAL
(1 / (4 * Z)) * (diff ((sin * x),f)) is V22() V23() ext-real Element of REAL
(sin * x) `| x is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
((sin * x) `| x) . f is V22() V23() ext-real Element of REAL
(1 / (4 * Z)) * (((sin * x) `| x) . f) is V22() V23() ext-real Element of REAL
((2 * Z) * f) + 0 is V22() V23() ext-real Element of REAL
cos . (((2 * Z) * f) + 0) is V22() V23() ext-real Element of REAL
(2 * Z) * (cos . (((2 * Z) * f) + 0)) is V22() V23() ext-real Element of REAL
(1 / (4 * Z)) * ((2 * Z) * (cos . (((2 * Z) * f) + 0))) is V22() V23() ext-real Element of REAL
(1 / (4 * Z)) * (2 * Z) is V22() V23() ext-real Element of REAL
((1 / (4 * Z)) * (2 * Z)) * (cos . (((2 * Z) * f) + 0)) is V22() V23() ext-real Element of REAL
1 / 4 is V22() V23() ext-real Element of REAL
1 / Z is V22() V23() ext-real Element of REAL
(1 / 4) * (1 / Z) is V22() V23() ext-real Element of REAL
((1 / 4) * (1 / Z)) * (2 * Z) is V22() V23() ext-real Element of REAL
cos . ((2 * Z) * f) is V22() V23() ext-real Element of REAL
(((1 / 4) * (1 / Z)) * (2 * Z)) * (cos . ((2 * Z) * f)) is V22() V23() ext-real Element of REAL
(1 / 4) * 2 is V22() V23() ext-real Element of REAL
(1 / Z) * Z is V22() V23() ext-real Element of REAL
((1 / 4) * 2) * ((1 / Z) * Z) is V22() V23() ext-real Element of REAL
(((1 / 4) * 2) * ((1 / Z) * Z)) * (cos . ((2 * Z) * f)) is V22() V23() ext-real Element of REAL
(1 / 2) * 1 is V22() V23() ext-real Element of REAL
((1 / 2) * 1) * (cos . ((2 * Z) * f)) is V22() V23() ext-real Element of REAL
f is V22() V23() ext-real Element of REAL
(((1 / (4 * Z)) (#) (sin * x)) `| x) . f is V22() V23() ext-real Element of REAL
(2 * Z) * f is V22() V23() ext-real Element of REAL
cos ((2 * Z) * f) is V22() V23() ext-real Element of REAL
(1 / 2) * (cos ((2 * Z) * f)) is V22() V23() ext-real Element of REAL
Z is V22() V23() ext-real Element of REAL
4 * Z is V22() V23() ext-real Element of REAL
1 / (4 * Z) is V22() V23() ext-real Element of REAL
2 * Z is V22() V23() ext-real Element of REAL
x is open V49() V50() V51() Element of K19(REAL)
x is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
f is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
sin * f is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
(1 / (4 * Z)) (#) (sin * f) is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
x - ((1 / (4 * Z)) (#) (sin * f)) is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
- ((1 / (4 * Z)) (#) (sin * f)) is Relation-like V6() V39() set
K58(1) is V22() V23() V68() set
K58(1) (#) ((1 / (4 * Z)) (#) (sin * f)) is Relation-like V6() set
x + (- ((1 / (4 * Z)) (#) (sin * f))) is Relation-like V6() set
dom (x - ((1 / (4 * Z)) (#) (sin * f))) is set
(x - ((1 / (4 * Z)) (#) (sin * f))) `| x is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
dom x is set
dom ((1 / (4 * Z)) (#) (sin * f)) is set
(dom x) /\ (dom ((1 / (4 * Z)) (#) (sin * f))) is set
x is V22() V23() ext-real Element of REAL
x . x is V22() V23() ext-real Element of REAL
(1 / 2) * x is V22() V23() ext-real Element of REAL
((1 / 2) * x) + 0 is V22() V23() ext-real Element of REAL
x / 2 is V22() V23() ext-real Element of REAL
x is V22() V23() ext-real Element of REAL
f . x is V22() V23() ext-real Element of REAL
(2 * Z) * x is V22() V23() ext-real Element of REAL
x is V22() V23() ext-real Element of REAL
((x - ((1 / (4 * Z)) (#) (sin * f))) `| x) . x is V22() V23() ext-real Element of REAL
Z * x is V22() V23() ext-real Element of REAL
sin (Z * x) is V22() V23() ext-real Element of REAL
(sin (Z * x)) ^2 is V22() V23() ext-real Element of REAL
K57((sin (Z * x)),(sin (Z * x))) is set
diff (x,x) is V22() V23() ext-real Element of REAL
diff (((1 / (4 * Z)) (#) (sin * f)),x) is V22() V23() ext-real Element of REAL
(diff (x,x)) - (diff (((1 / (4 * Z)) (#) (sin * f)),x)) is V22() V23() ext-real Element of REAL
x `| x is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
(x `| x) . x is V22() V23() ext-real Element of REAL
((x `| x) . x) - (diff (((1 / (4 * Z)) (#) (sin * f)),x)) is V22() V23() ext-real Element of REAL
((1 / (4 * Z)) (#) (sin * f)) `| x is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
(((1 / (4 * Z)) (#) (sin * f)) `| x) . x is V22() V23() ext-real Element of REAL
((x `| x) . x) - ((((1 / (4 * Z)) (#) (sin * f)) `| x) . x) is V22() V23() ext-real Element of REAL
(2 * Z) * x is V22() V23() ext-real Element of REAL
cos ((2 * Z) * x) is V22() V23() ext-real Element of REAL
(1 / 2) * (cos ((2 * Z) * x)) is V22() V23() ext-real Element of REAL
((x `| x) . x) - ((1 / 2) * (cos ((2 * Z) * x))) is V22() V23() ext-real Element of REAL
(1 / 2) - ((1 / 2) * (cos ((2 * Z) * x))) is V22() V23() ext-real Element of REAL
2 * (Z * x) is V22() V23() ext-real Element of REAL
cos (2 * (Z * x)) is V22() V23() ext-real Element of REAL
1 - (cos (2 * (Z * x))) is V22() V23() ext-real Element of REAL
(1 / 2) * (1 - (cos (2 * (Z * x)))) is V22() V23() ext-real Element of REAL
2 * ((sin (Z * x)) ^2) is V22() V23() ext-real Element of REAL
(1 / 2) * (2 * ((sin (Z * x)) ^2)) is V22() V23() ext-real Element of REAL
x is V22() V23() ext-real Element of REAL
((x - ((1 / (4 * Z)) (#) (sin * f))) `| x) . x is V22() V23() ext-real Element of REAL
Z * x is V22() V23() ext-real Element of REAL
sin (Z * x) is V22() V23() ext-real Element of REAL
(sin (Z * x)) ^2 is V22() V23() ext-real Element of REAL
K57((sin (Z * x)),(sin (Z * x))) is set
Z is V22() V23() ext-real Element of REAL
4 * Z is V22() V23() ext-real Element of REAL
1 / (4 * Z) is V22() V23() ext-real Element of REAL
2 * Z is V22() V23() ext-real Element of REAL
x is open V49() V50() V51() Element of K19(REAL)
x is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
f is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
sin * f is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
(1 / (4 * Z)) (#) (sin * f) is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
x + ((1 / (4 * Z)) (#) (sin * f)) is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
dom (x + ((1 / (4 * Z)) (#) (sin * f))) is set
(x + ((1 / (4 * Z)) (#) (sin * f))) `| x is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
dom x is set
dom ((1 / (4 * Z)) (#) (sin * f)) is set
(dom x) /\ (dom ((1 / (4 * Z)) (#) (sin * f))) is set
x is V22() V23() ext-real Element of REAL
x . x is V22() V23() ext-real Element of REAL
(1 / 2) * x is V22() V23() ext-real Element of REAL
((1 / 2) * x) + 0 is V22() V23() ext-real Element of REAL
x / 2 is V22() V23() ext-real Element of REAL
x is V22() V23() ext-real Element of REAL
f . x is V22() V23() ext-real Element of REAL
(2 * Z) * x is V22() V23() ext-real Element of REAL
x is V22() V23() ext-real Element of REAL
((x + ((1 / (4 * Z)) (#) (sin * f))) `| x) . x is V22() V23() ext-real Element of REAL
Z * x is V22() V23() ext-real Element of REAL
cos (Z * x) is V22() V23() ext-real Element of REAL
(cos (Z * x)) ^2 is V22() V23() ext-real Element of REAL
K57((cos (Z * x)),(cos (Z * x))) is set
diff (x,x) is V22() V23() ext-real Element of REAL
diff (((1 / (4 * Z)) (#) (sin * f)),x) is V22() V23() ext-real Element of REAL
(diff (x,x)) + (diff (((1 / (4 * Z)) (#) (sin * f)),x)) is V22() V23() ext-real Element of REAL
x `| x is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
(x `| x) . x is V22() V23() ext-real Element of REAL
((x `| x) . x) + (diff (((1 / (4 * Z)) (#) (sin * f)),x)) is V22() V23() ext-real Element of REAL
((1 / (4 * Z)) (#) (sin * f)) `| x is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
(((1 / (4 * Z)) (#) (sin * f)) `| x) . x is V22() V23() ext-real Element of REAL
((x `| x) . x) + ((((1 / (4 * Z)) (#) (sin * f)) `| x) . x) is V22() V23() ext-real Element of REAL
(2 * Z) * x is V22() V23() ext-real Element of REAL
cos ((2 * Z) * x) is V22() V23() ext-real Element of REAL
(1 / 2) * (cos ((2 * Z) * x)) is V22() V23() ext-real Element of REAL
((x `| x) . x) + ((1 / 2) * (cos ((2 * Z) * x))) is V22() V23() ext-real Element of REAL
(1 / 2) + ((1 / 2) * (cos ((2 * Z) * x))) is V22() V23() ext-real Element of REAL
2 * (Z * x) is V22() V23() ext-real Element of REAL
cos (2 * (Z * x)) is V22() V23() ext-real Element of REAL
1 + (cos (2 * (Z * x))) is V22() V23() ext-real Element of REAL
(1 / 2) * (1 + (cos (2 * (Z * x)))) is V22() V23() ext-real Element of REAL
2 * ((cos (Z * x)) ^2) is V22() V23() ext-real Element of REAL
(1 / 2) * (2 * ((cos (Z * x)) ^2)) is V22() V23() ext-real Element of REAL
x is V22() V23() ext-real Element of REAL
((x + ((1 / (4 * Z)) (#) (sin * f))) `| x) . x is V22() V23() ext-real Element of REAL
Z * x is V22() V23() ext-real Element of REAL
cos (Z * x) is V22() V23() ext-real Element of REAL
(cos (Z * x)) ^2 is V22() V23() ext-real Element of REAL
K57((cos (Z * x)),(cos (Z * x))) is set
Z is V15() V16() V17() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() Element of NAT
#Z Z is Relation-like REAL -defined REAL -valued V6() V30( REAL , REAL ) V39() V40() V41() Element of K19(K20(REAL,REAL))
(#Z Z) * cos is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
1 / Z is V22() V23() ext-real Element of REAL
(1 / Z) (#) ((#Z Z) * cos) is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
dom ((1 / Z) (#) ((#Z Z) * cos)) is set
Z - 1 is V22() V23() ext-real V68() Element of REAL
x is open V49() V50() V51() Element of K19(REAL)
((1 / Z) (#) ((#Z Z) * cos)) `| x is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
x is V22() V23() ext-real Element of REAL
dom ((#Z Z) * cos) is set
x is V22() V23() ext-real Element of REAL
(((1 / Z) (#) ((#Z Z) * cos)) `| x) . x is V22() V23() ext-real Element of REAL
cos . x is V22() V23() ext-real Element of REAL
(cos . x) #Z (Z - 1) is V22() V23() ext-real Element of REAL
sin . x is V22() V23() ext-real Element of REAL
((cos . x) #Z (Z - 1)) * (sin . x) is V22() V23() ext-real Element of REAL
- (((cos . x) #Z (Z - 1)) * (sin . x)) is V22() V23() ext-real Element of REAL
diff (((#Z Z) * cos),x) is V22() V23() ext-real Element of REAL
(1 / Z) * (diff (((#Z Z) * cos),x)) is V22() V23() ext-real Element of REAL
Z * ((cos . x) #Z (Z - 1)) is V22() V23() ext-real Element of REAL
diff (cos,x) is V22() V23() ext-real Element of REAL
(Z * ((cos . x) #Z (Z - 1))) * (diff (cos,x)) is V22() V23() ext-real Element of REAL
(1 / Z) * ((Z * ((cos . x) #Z (Z - 1))) * (diff (cos,x))) is V22() V23() ext-real Element of REAL
- (sin . x) is V22() V23() ext-real Element of REAL
(Z * ((cos . x) #Z (Z - 1))) * (- (sin . x)) is V22() V23() ext-real Element of REAL
(1 / Z) * ((Z * ((cos . x) #Z (Z - 1))) * (- (sin . x))) is V22() V23() ext-real Element of REAL
(1 / Z) * Z is V22() V23() ext-real Element of REAL
((1 / Z) * Z) * ((cos . x) #Z (Z - 1)) is V22() V23() ext-real Element of REAL
(((1 / Z) * Z) * ((cos . x) #Z (Z - 1))) * (- (sin . x)) is V22() V23() ext-real Element of REAL
Z " is V22() V23() ext-real Element of REAL
(Z ") * Z is V22() V23() ext-real Element of REAL
((Z ") * Z) * ((cos . x) #Z (Z - 1)) is V22() V23() ext-real Element of REAL
(((Z ") * Z) * ((cos . x) #Z (Z - 1))) * (- (sin . x)) is V22() V23() ext-real Element of REAL
1 * ((cos . x) #Z (Z - 1)) is V22() V23() ext-real Element of REAL
(1 * ((cos . x) #Z (Z - 1))) * (- (sin . x)) is V22() V23() ext-real Element of REAL
x is V22() V23() ext-real Element of REAL
(((1 / Z) (#) ((#Z Z) * cos)) `| x) . x is V22() V23() ext-real Element of REAL
cos . x is V22() V23() ext-real Element of REAL
(cos . x) #Z (Z - 1) is V22() V23() ext-real Element of REAL
sin . x is V22() V23() ext-real Element of REAL
((cos . x) #Z (Z - 1)) * (sin . x) is V22() V23() ext-real Element of REAL
- (((cos . x) #Z (Z - 1)) * (sin . x)) is V22() V23() ext-real Element of REAL
#Z 3 is Relation-like REAL -defined REAL -valued V6() V30( REAL , REAL ) V39() V40() V41() Element of K19(K20(REAL,REAL))
(#Z 3) * cos is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
1 / 3 is V22() V23() ext-real Element of REAL
(1 / 3) (#) ((#Z 3) * cos) is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
((1 / 3) (#) ((#Z 3) * cos)) - cos is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
- cos is Relation-like V6() V39() set
K58(1) is V22() V23() V68() set
K58(1) (#) cos is Relation-like V6() set
((1 / 3) (#) ((#Z 3) * cos)) + (- cos) is Relation-like V6() set
dom (((1 / 3) (#) ((#Z 3) * cos)) - cos) is set
Z is open V49() V50() V51() Element of K19(REAL)
(((1 / 3) (#) ((#Z 3) * cos)) - cos) `| Z is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
dom ((1 / 3) (#) ((#Z 3) * cos)) is set
dom cos is set
(dom ((1 / 3) (#) ((#Z 3) * cos))) /\ (dom cos) is set
x is V22() V23() ext-real Element of REAL
((((1 / 3) (#) ((#Z 3) * cos)) - cos) `| Z) . x is V22() V23() ext-real Element of REAL
diff (((1 / 3) (#) ((#Z 3) * cos)),x) is V22() V23() ext-real Element of REAL
diff (cos,x) is V22() V23() ext-real Element of REAL
(diff (((1 / 3) (#) ((#Z 3) * cos)),x)) - (diff (cos,x)) is V22() V23() ext-real Element of REAL
sin . x is V22() V23() ext-real Element of REAL
- (sin . x) is V22() V23() ext-real Element of REAL
(diff (((1 / 3) (#) ((#Z 3) * cos)),x)) - (- (sin . x)) is V22() V23() ext-real Element of REAL
((1 / 3) (#) ((#Z 3) * cos)) `| Z is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
(((1 / 3) (#) ((#Z 3) * cos)) `| Z) . x is V22() V23() ext-real Element of REAL
((((1 / 3) (#) ((#Z 3) * cos)) `| Z) . x) - (- (sin . x)) is V22() V23() ext-real Element of REAL
cos . x is V22() V23() ext-real Element of REAL
3 - 1 is V22() V23() ext-real V68() Element of REAL
(cos . x) #Z (3 - 1) is V22() V23() ext-real Element of REAL
((cos . x) #Z (3 - 1)) * (sin . x) is V22() V23() ext-real Element of REAL
- (((cos . x) #Z (3 - 1)) * (sin . x)) is V22() V23() ext-real Element of REAL
(- (((cos . x) #Z (3 - 1)) * (sin . x))) - (- (sin . x)) is V22() V23() ext-real Element of REAL
(cos . x) #Z 2 is V22() V23() ext-real Element of REAL
1 - ((cos . x) #Z 2) is V22() V23() ext-real Element of REAL
(sin . x) * (1 - ((cos . x) #Z 2)) is V22() V23() ext-real Element of REAL
abs 2 is V15() V16() V17() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() Element of NAT
(cos . x) |^ (abs 2) is V22() V23() ext-real Element of REAL
1 - ((cos . x) |^ (abs 2)) is V22() V23() ext-real Element of REAL
(sin . x) * (1 - ((cos . x) |^ (abs 2))) is V22() V23() ext-real Element of REAL
(cos . x) |^ 2 is V22() V23() ext-real Element of REAL
1 - ((cos . x) |^ 2) is V22() V23() ext-real Element of REAL
(sin . x) * (1 - ((cos . x) |^ 2)) is V22() V23() ext-real Element of REAL
(cos . x) * (cos . x) is V22() V23() ext-real Element of REAL
1 - ((cos . x) * (cos . x)) is V22() V23() ext-real Element of REAL
(sin . x) * (1 - ((cos . x) * (cos . x))) is V22() V23() ext-real Element of REAL
(sin . x) * (sin . x) is V22() V23() ext-real Element of REAL
((cos . x) * (cos . x)) + ((sin . x) * (sin . x)) is V22() V23() ext-real Element of REAL
(((cos . x) * (cos . x)) + ((sin . x) * (sin . x))) - ((cos . x) * (cos . x)) is V22() V23() ext-real Element of REAL
(sin . x) * ((((cos . x) * (cos . x)) + ((sin . x) * (sin . x))) - ((cos . x) * (cos . x))) is V22() V23() ext-real Element of REAL
(sin . x) |^ 2 is V22() V23() ext-real Element of REAL
(sin . x) * ((sin . x) |^ 2) is V22() V23() ext-real Element of REAL
2 + 1 is V15() V16() V17() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() Element of NAT
(sin . x) |^ (2 + 1) is V22() V23() ext-real Element of REAL
(sin . x) |^ 3 is V22() V23() ext-real Element of REAL
x is V22() V23() ext-real Element of REAL
((((1 / 3) (#) ((#Z 3) * cos)) - cos) `| Z) . x is V22() V23() ext-real Element of REAL
sin . x is V22() V23() ext-real Element of REAL
(sin . x) |^ 3 is V22() V23() ext-real Element of REAL
(#Z 3) * sin is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
(1 / 3) (#) ((#Z 3) * sin) is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
sin - ((1 / 3) (#) ((#Z 3) * sin)) is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
- ((1 / 3) (#) ((#Z 3) * sin)) is Relation-like V6() V39() set
K58(1) (#) ((1 / 3) (#) ((#Z 3) * sin)) is Relation-like V6() set
sin + (- ((1 / 3) (#) ((#Z 3) * sin))) is Relation-like V6() set
dom (sin - ((1 / 3) (#) ((#Z 3) * sin))) is set
Z is open V49() V50() V51() Element of K19(REAL)
(sin - ((1 / 3) (#) ((#Z 3) * sin))) `| Z is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
dom ((1 / 3) (#) ((#Z 3) * sin)) is set
dom sin is set
(dom ((1 / 3) (#) ((#Z 3) * sin))) /\ (dom sin) is set
x is V22() V23() ext-real Element of REAL
((sin - ((1 / 3) (#) ((#Z 3) * sin))) `| Z) . x is V22() V23() ext-real Element of REAL
diff (sin,x) is V22() V23() ext-real Element of REAL
diff (((1 / 3) (#) ((#Z 3) * sin)),x) is V22() V23() ext-real Element of REAL
(diff (sin,x)) - (diff (((1 / 3) (#) ((#Z 3) * sin)),x)) is V22() V23() ext-real Element of REAL
cos . x is V22() V23() ext-real Element of REAL
(cos . x) - (diff (((1 / 3) (#) ((#Z 3) * sin)),x)) is V22() V23() ext-real Element of REAL
((1 / 3) (#) ((#Z 3) * sin)) `| Z is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
(((1 / 3) (#) ((#Z 3) * sin)) `| Z) . x is V22() V23() ext-real Element of REAL
(cos . x) - ((((1 / 3) (#) ((#Z 3) * sin)) `| Z) . x) is V22() V23() ext-real Element of REAL
sin . x is V22() V23() ext-real Element of REAL
3 - 1 is V22() V23() ext-real V68() Element of REAL
(sin . x) #Z (3 - 1) is V22() V23() ext-real Element of REAL
((sin . x) #Z (3 - 1)) * (cos . x) is V22() V23() ext-real Element of REAL
(cos . x) - (((sin . x) #Z (3 - 1)) * (cos . x)) is V22() V23() ext-real Element of REAL
(sin . x) #Z 2 is V22() V23() ext-real Element of REAL
1 - ((sin . x) #Z 2) is V22() V23() ext-real Element of REAL
(cos . x) * (1 - ((sin . x) #Z 2)) is V22() V23() ext-real Element of REAL
abs 2 is V15() V16() V17() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() Element of NAT
(sin . x) |^ (abs 2) is V22() V23() ext-real Element of REAL
1 - ((sin . x) |^ (abs 2)) is V22() V23() ext-real Element of REAL
(cos . x) * (1 - ((sin . x) |^ (abs 2))) is V22() V23() ext-real Element of REAL
(sin . x) |^ 2 is V22() V23() ext-real Element of REAL
1 - ((sin . x) |^ 2) is V22() V23() ext-real Element of REAL
(cos . x) * (1 - ((sin . x) |^ 2)) is V22() V23() ext-real Element of REAL
(sin . x) * (sin . x) is V22() V23() ext-real Element of REAL
1 - ((sin . x) * (sin . x)) is V22() V23() ext-real Element of REAL
(cos . x) * (1 - ((sin . x) * (sin . x))) is V22() V23() ext-real Element of REAL
(cos . x) * (cos . x) is V22() V23() ext-real Element of REAL
((cos . x) * (cos . x)) + ((sin . x) * (sin . x)) is V22() V23() ext-real Element of REAL
(((cos . x) * (cos . x)) + ((sin . x) * (sin . x))) - ((sin . x) * (sin . x)) is V22() V23() ext-real Element of REAL
(cos . x) * ((((cos . x) * (cos . x)) + ((sin . x) * (sin . x))) - ((sin . x) * (sin . x))) is V22() V23() ext-real Element of REAL
(cos . x) |^ 2 is V22() V23() ext-real Element of REAL
(cos . x) * ((cos . x) |^ 2) is V22() V23() ext-real Element of REAL
2 + 1 is V15() V16() V17() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() Element of NAT
(cos . x) |^ (2 + 1) is V22() V23() ext-real Element of REAL
(cos . x) |^ 3 is V22() V23() ext-real Element of REAL
x is V22() V23() ext-real Element of REAL
((sin - ((1 / 3) (#) ((#Z 3) * sin))) `| Z) . x is V22() V23() ext-real Element of REAL
cos . x is V22() V23() ext-real Element of REAL
(cos . x) |^ 3 is V22() V23() ext-real Element of REAL
sin * ln is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
dom (sin * ln) is set
Z is open V49() V50() V51() Element of K19(REAL)
(sin * ln) `| Z is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
dom ln is set
x is set
x is V22() V23() ext-real Element of REAL
ln . x is V22() V23() ext-real Element of REAL
x is V22() V23() ext-real Element of REAL
((sin * ln) `| Z) . x is V22() V23() ext-real Element of REAL
log (number_e,x) is V22() V23() ext-real Element of REAL
cos . (log (number_e,x)) is V22() V23() ext-real Element of REAL
(cos . (log (number_e,x))) / x is V22() V23() ext-real Element of REAL
ln . x is V22() V23() ext-real Element of REAL
diff ((sin * ln),x) is V22() V23() ext-real Element of REAL
diff (sin,(ln . x)) is V22() V23() ext-real Element of REAL
diff (ln,x) is V22() V23() ext-real Element of REAL
(diff (sin,(ln . x))) * (diff (ln,x)) is V22() V23() ext-real Element of REAL
cos . (ln . x) is V22() V23() ext-real Element of REAL
(cos . (ln . x)) * (diff (ln,x)) is V22() V23() ext-real Element of REAL
(cos . (log (number_e,x))) * (diff (ln,x)) is V22() V23() ext-real Element of REAL
1 / x is V22() V23() ext-real Element of REAL
(cos . (log (number_e,x))) * (1 / x) is V22() V23() ext-real Element of REAL
x is V22() V23() ext-real Element of REAL
((sin * ln) `| Z) . x is V22() V23() ext-real Element of REAL
log (number_e,x) is V22() V23() ext-real Element of REAL
cos . (log (number_e,x)) is V22() V23() ext-real Element of REAL
(cos . (log (number_e,x))) / x is V22() V23() ext-real Element of REAL
cos * ln is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
- (cos * ln) is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
K58(1) (#) (cos * ln) is Relation-like V6() set
dom (- (cos * ln)) is set
Z is open V49() V50() V51() Element of K19(REAL)
(- (cos * ln)) `| Z is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
dom (cos * ln) is set
dom ln is set
x is set
x is V22() V23() ext-real Element of REAL
ln . x is V22() V23() ext-real Element of REAL
x is V22() V23() ext-real Element of REAL
((- (cos * ln)) `| Z) . x is V22() V23() ext-real Element of REAL
log (number_e,x) is V22() V23() ext-real Element of REAL
sin . (log (number_e,x)) is V22() V23() ext-real Element of REAL
(sin . (log (number_e,x))) / x is V22() V23() ext-real Element of REAL
ln . x is V22() V23() ext-real Element of REAL
diff ((cos * ln),x) is V22() V23() ext-real Element of REAL
(- 1) * (diff ((cos * ln),x)) is V22() V23() ext-real Element of REAL
diff (cos,(ln . x)) is V22() V23() ext-real Element of REAL
diff (ln,x) is V22() V23() ext-real Element of REAL
(diff (cos,(ln . x))) * (diff (ln,x)) is V22() V23() ext-real Element of REAL
(- 1) * ((diff (cos,(ln . x))) * (diff (ln,x))) is V22() V23() ext-real Element of REAL
sin . (ln . x) is V22() V23() ext-real Element of REAL
- (sin . (ln . x)) is V22() V23() ext-real Element of REAL
(- (sin . (ln . x))) * (diff (ln,x)) is V22() V23() ext-real Element of REAL
(- 1) * ((- (sin . (ln . x))) * (diff (ln,x))) is V22() V23() ext-real Element of REAL
(- 1) * (- (sin . (ln . x))) is V22() V23() ext-real Element of REAL
((- 1) * (- (sin . (ln . x)))) * (diff (ln,x)) is V22() V23() ext-real Element of REAL
- (sin . (log (number_e,x))) is V22() V23() ext-real Element of REAL
(- 1) * (- (sin . (log (number_e,x)))) is V22() V23() ext-real Element of REAL
((- 1) * (- (sin . (log (number_e,x))))) * (diff (ln,x)) is V22() V23() ext-real Element of REAL
1 / x is V22() V23() ext-real Element of REAL
((- 1) * (- (sin . (log (number_e,x))))) * (1 / x) is V22() V23() ext-real Element of REAL
(- 1) (#) (cos * ln) is Relation-like REAL -defined REAL -valued V6() V39() V40() V41() Element of K19(K20(REAL,REAL))
dom ((- 1) (#) (cos * ln)) is set
x is V22() V23() ext-real Element of REAL
((- (cos * ln)) `| Z) . x is V22() V23() ext-real Element of REAL
log (number_e,x) is V22() V23() ext-real Element of REAL
sin . (log (number_e,x)) is V22() V23() ext-real Element of REAL
(sin . (log (number_e,x))) / x is V22() V23() ext-real Element of REAL