let n be Element of NAT ; for Z being open Subset of REAL st Z c= dom ((- (1 / n)) (#) ((#Z n) * (sin ^))) & n > 0 & ( for x being Real st x in Z holds
sin . x <> 0 ) holds
( (- (1 / n)) (#) ((#Z n) * (sin ^)) is_differentiable_on Z & ( for x being Real st x in Z holds
(((- (1 / n)) (#) ((#Z n) * (sin ^))) `| Z) . x = (cos . x) / ((sin . x) #Z (n + 1)) ) )
let Z be open Subset of REAL; ( Z c= dom ((- (1 / n)) (#) ((#Z n) * (sin ^))) & n > 0 & ( for x being Real st x in Z holds
sin . x <> 0 ) implies ( (- (1 / n)) (#) ((#Z n) * (sin ^)) is_differentiable_on Z & ( for x being Real st x in Z holds
(((- (1 / n)) (#) ((#Z n) * (sin ^))) `| Z) . x = (cos . x) / ((sin . x) #Z (n + 1)) ) ) )
assume that
A1:
Z c= dom ((- (1 / n)) (#) ((#Z n) * (sin ^)))
and
A2:
n > 0
and
A3:
for x being Real st x in Z holds
sin . x <> 0
; ( (- (1 / n)) (#) ((#Z n) * (sin ^)) is_differentiable_on Z & ( for x being Real st x in Z holds
(((- (1 / n)) (#) ((#Z n) * (sin ^))) `| Z) . x = (cos . x) / ((sin . x) #Z (n + 1)) ) )
A4:
Z c= dom ((#Z n) * (sin ^))
by A1, VALUED_1:def 5;
A5:
sin ^ is_differentiable_on Z
by A3, FDIFF_4:40;
then A6:
(#Z n) * (sin ^) is_differentiable_on Z
by A4, FDIFF_1:9;
for y being object st y in Z holds
y in dom (sin ^)
by A4, FUNCT_1:11;
then A7:
Z c= dom (sin ^)
by TARSKI:def 3;
for x being Real st x in Z holds
(((- (1 / n)) (#) ((#Z n) * (sin ^))) `| Z) . x = (cos . x) / ((sin . x) #Z (n + 1))
proof
let x be
Real;
( x in Z implies (((- (1 / n)) (#) ((#Z n) * (sin ^))) `| Z) . x = (cos . x) / ((sin . x) #Z (n + 1)) )
assume A8:
x in Z
;
(((- (1 / n)) (#) ((#Z n) * (sin ^))) `| Z) . x = (cos . x) / ((sin . x) #Z (n + 1))
then A9:
sin ^ is_differentiable_in x
by A5, FDIFF_1:9;
A10:
(sin ^) . x =
(sin . x) "
by A7, A8, RFUNCT_1:def 2
.=
1
/ (sin . x)
by XCMPLX_1:215
;
(((- (1 / n)) (#) ((#Z n) * (sin ^))) `| Z) . x =
(- (1 / n)) * (diff (((#Z n) * (sin ^)),x))
by A1, A6, A8, FDIFF_1:20
.=
(- (1 / n)) * ((n * (((sin ^) . x) #Z (n - 1))) * (diff ((sin ^),x)))
by A9, TAYLOR_1:3
.=
(- (1 / n)) * ((n * (((sin ^) . x) #Z (n - 1))) * (((sin ^) `| Z) . x))
by A5, A8, FDIFF_1:def 7
.=
(- (1 / n)) * ((n * (((sin ^) . x) #Z (n - 1))) * (- ((cos . x) / ((sin . x) ^2))))
by A3, A8, FDIFF_4:40
.=
- ((((1 / n) * n) * (((sin ^) . x) #Z (n - 1))) * (- ((cos . x) / ((sin . x) ^2))))
.=
- ((1 * (((sin ^) . x) #Z (n - 1))) * (- ((cos . x) / ((sin . x) ^2))))
by A2, XCMPLX_1:106
.=
- (((1 / (sin . x)) #Z (n - 1)) * (- ((cos . x) / ((sin . x) #Z 2))))
by A10, Th1
.=
- ((- ((cos . x) / ((sin . x) #Z 2))) * (1 / ((sin . x) #Z (n - 1))))
by PREPOWER:42
.=
((cos . x) / ((sin . x) #Z 2)) * (1 / ((sin . x) #Z (n - 1)))
.=
((cos . x) / ((sin . x) #Z 2)) / ((sin . x) #Z (n - 1))
by XCMPLX_1:99
.=
(cos . x) / (((sin . x) #Z 2) * ((sin . x) #Z (n - 1)))
by XCMPLX_1:78
.=
(cos . x) / ((sin . x) #Z (2 + (n - 1)))
by A3, A8, PREPOWER:44
.=
(cos . x) / ((sin . x) #Z (n + 1))
;
hence
(((- (1 / n)) (#) ((#Z n) * (sin ^))) `| Z) . x = (cos . x) / ((sin . x) #Z (n + 1))
;
verum
end;
hence
( (- (1 / n)) (#) ((#Z n) * (sin ^)) is_differentiable_on Z & ( for x being Real st x in Z holds
(((- (1 / n)) (#) ((#Z n) * (sin ^))) `| Z) . x = (cos . x) / ((sin . x) #Z (n + 1)) ) )
by A1, A6, FDIFF_1:20; verum