let n be Element of NAT ; for Z being open Subset of REAL st Z c= dom ((1 / n) (#) ((#Z n) * (cos ^))) & n > 0 & ( for x being Real st x in Z holds
cos . x <> 0 ) holds
( (1 / n) (#) ((#Z n) * (cos ^)) is_differentiable_on Z & ( for x being Real st x in Z holds
(((1 / n) (#) ((#Z n) * (cos ^))) `| Z) . x = (sin . x) / ((cos . x) #Z (n + 1)) ) )
let Z be open Subset of REAL; ( Z c= dom ((1 / n) (#) ((#Z n) * (cos ^))) & n > 0 & ( for x being Real st x in Z holds
cos . x <> 0 ) implies ( (1 / n) (#) ((#Z n) * (cos ^)) is_differentiable_on Z & ( for x being Real st x in Z holds
(((1 / n) (#) ((#Z n) * (cos ^))) `| Z) . x = (sin . x) / ((cos . x) #Z (n + 1)) ) ) )
assume that
A1:
Z c= dom ((1 / n) (#) ((#Z n) * (cos ^)))
and
A2:
n > 0
and
A3:
for x being Real st x in Z holds
cos . x <> 0
; ( (1 / n) (#) ((#Z n) * (cos ^)) is_differentiable_on Z & ( for x being Real st x in Z holds
(((1 / n) (#) ((#Z n) * (cos ^))) `| Z) . x = (sin . x) / ((cos . x) #Z (n + 1)) ) )
A4:
Z c= dom ((#Z n) * (cos ^))
by A1, VALUED_1:def 5;
A5:
cos ^ is_differentiable_on Z
by A3, FDIFF_4:39;
then A6:
(#Z n) * (cos ^) is_differentiable_on Z
by A4, FDIFF_1:9;
for y being object st y in Z holds
y in dom (cos ^)
by A4, FUNCT_1:11;
then A7:
Z c= dom (cos ^)
by TARSKI:def 3;
for x being Real st x in Z holds
(((1 / n) (#) ((#Z n) * (cos ^))) `| Z) . x = (sin . x) / ((cos . x) #Z (n + 1))
proof
let x be
Real;
( x in Z implies (((1 / n) (#) ((#Z n) * (cos ^))) `| Z) . x = (sin . x) / ((cos . x) #Z (n + 1)) )
assume A8:
x in Z
;
(((1 / n) (#) ((#Z n) * (cos ^))) `| Z) . x = (sin . x) / ((cos . x) #Z (n + 1))
then A9:
cos ^ is_differentiable_in x
by A5, FDIFF_1:9;
A10:
(cos ^) . x =
(cos . x) "
by A7, A8, RFUNCT_1:def 2
.=
1
/ (cos . x)
by XCMPLX_1:215
;
(((1 / n) (#) ((#Z n) * (cos ^))) `| Z) . x =
(1 / n) * (diff (((#Z n) * (cos ^)),x))
by A1, A6, A8, FDIFF_1:20
.=
(1 / n) * ((n * (((cos ^) . x) #Z (n - 1))) * (diff ((cos ^),x)))
by A9, TAYLOR_1:3
.=
(1 / n) * ((n * (((cos ^) . x) #Z (n - 1))) * (((cos ^) `| Z) . x))
by A5, A8, FDIFF_1:def 7
.=
(1 / n) * ((n * (((cos ^) . x) #Z (n - 1))) * ((sin . x) / ((cos . x) ^2)))
by A3, A8, FDIFF_4:39
.=
(((1 / n) * n) * (((cos ^) . x) #Z (n - 1))) * ((sin . x) / ((cos . x) ^2))
.=
(1 * (((cos ^) . x) #Z (n - 1))) * ((sin . x) / ((cos . x) ^2))
by A2, XCMPLX_1:106
.=
((1 / (cos . x)) #Z (n - 1)) * ((sin . x) / ((cos . x) #Z 2))
by A10, Th1
.=
(1 / ((cos . x) #Z (n - 1))) * ((sin . x) / ((cos . x) #Z 2))
by PREPOWER:42
.=
((sin . x) / ((cos . x) #Z 2)) / ((cos . x) #Z (n - 1))
by XCMPLX_1:99
.=
(sin . x) / (((cos . x) #Z 2) * ((cos . x) #Z (n - 1)))
by XCMPLX_1:78
.=
(sin . x) / ((cos . x) #Z (2 + (n - 1)))
by A3, A8, PREPOWER:44
.=
(sin . x) / ((cos . x) #Z (n + 1))
;
hence
(((1 / n) (#) ((#Z n) * (cos ^))) `| Z) . x = (sin . x) / ((cos . x) #Z (n + 1))
;
verum
end;
hence
( (1 / n) (#) ((#Z n) * (cos ^)) is_differentiable_on Z & ( for x being Real st x in Z holds
(((1 / n) (#) ((#Z n) * (cos ^))) `| Z) . x = (sin . x) / ((cos . x) #Z (n + 1)) ) )
by A1, A6, FDIFF_1:20; verum