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(