let n be Element of NAT ; :: thesis: for Z being open Subset of REAL
for f being PartFunc of REAL ,REAL st Z c= dom ((cos * f) (#) ((#Z n) * sin )) & n >= 1 & ( for x being Real st x in Z holds
f . x = n * x ) holds
( (cos * f) (#) ((#Z n) * sin ) is_differentiable_on Z & ( for x being Real st x in Z holds
(((cos * f) (#) ((#Z n) * sin )) `| Z) . x = (n * ((sin . x) #Z (n - 1))) * (cos . ((n + 1) * x)) ) )
let Z be open Subset of REAL ; :: thesis: for f being PartFunc of REAL ,REAL st Z c= dom ((cos * f) (#) ((#Z n) * sin )) & n >= 1 & ( for x being Real st x in Z holds
f . x = n * x ) holds
( (cos * f) (#) ((#Z n) * sin ) is_differentiable_on Z & ( for x being Real st x in Z holds
(((cos * f) (#) ((#Z n) * sin )) `| Z) . x = (n * ((sin . x) #Z (n - 1))) * (cos . ((n + 1) * x)) ) )
let f be PartFunc of REAL ,REAL ; :: thesis: ( Z c= dom ((cos * f) (#) ((#Z n) * sin )) & n >= 1 & ( for x being Real st x in Z holds
f . x = n * x ) implies ( (cos * f) (#) ((#Z n) * sin ) is_differentiable_on Z & ( for x being Real st x in Z holds
(((cos * f) (#) ((#Z n) * sin )) `| Z) . x = (n * ((sin . x) #Z (n - 1))) * (cos . ((n + 1) * x)) ) ) )
assume A1: ( Z c= dom ((cos * f) (#) ((#Z n) * sin )) & n >= 1 & ( for x being Real st x in Z holds
f . x = n * x ) ) ; :: thesis: ( (cos * f) (#) ((#Z n) * sin ) is_differentiable_on Z & ( for x being Real st x in Z holds
(((cos * f) (#) ((#Z n) * sin )) `| Z) . x = (n * ((sin . x) #Z (n - 1))) * (cos . ((n + 1) * x)) ) )
then Z c= (dom (cos * f)) /\ (dom ((#Z n) * sin )) by VALUED_1:def 4;
then A2: ( Z c= dom (cos * f) & Z c= dom ((#Z n) * sin ) ) by XBOOLE_1:18;
then for y being set st y in Z holds
y in dom f by FUNCT_1:21;
then A3: Z c= dom f by TARSKI:def 3;
A4: for x being Real st x in Z holds
f . x = (n * x) + 0 by A1;
then A5: ( f is_differentiable_on Z & ( for x being Real st x in Z holds
(f `| Z) . x = n ) ) by A3, FDIFF_1:31;
for x being Real st x in Z holds
cos * f is_differentiable_in x

proof

let x be Real; :: thesis: ( x in Z implies cos * f is_differentiable_in x )
assume x in Z ; :: thesis: cos * f is_differentiable_in x
then A6: f is_differentiable_in x by A5, FDIFF_1:16;
cos is_differentiable_in f . x by SIN_COS:68;
hence cos * f is_differentiable_in x by A6, FDIFF_2:13; :: thesis: verum

end;

then A7: cos * f is_differentiable_on Z by A2, FDIFF_1:16;
A8: for x being Real st x in Z holds
((cos * f) `| Z) . x = - (n * (sin . (n * x)))

proof

let x be Real; :: thesis: ( x in Z implies ((cos * f) `| Z) . x = - (n * (sin . (n * x))) )
assume A9: x in Z ; :: thesis: ((cos * f) `| Z) . x = - (n * (sin . (n * x)))
then A10: f is_differentiable_in x by A5, FDIFF_1:16;
cos is_differentiable_in f . x by SIN_COS:68;
then diff (cos * f),x = (diff cos ,(f . x)) * (diff f,x) by A10, FDIFF_2:13
.= (- (sin . (f . x))) * (diff f,x) by SIN_COS:68
.= (- (sin . (n * x))) * (diff f,x) by A1, A9
.= (- (sin . (n * x))) * ((f `| Z) . x) by A5, A9, FDIFF_1:def 8
.= (- (sin . (n * x))) * n by A3, A4, A9, FDIFF_1:31
.= - (n * (sin . (n * x))) ;
hence ((cos * f) `| Z) . x = - (n * (sin . (n * x))) by A7, A9, FDIFF_1:def 8; :: thesis: verum

end;

now

let x be Real; :: thesis: ( x in Z implies (#Z n) * sin is_differentiable_in x )
assume x in Z ; :: thesis: (#Z n) * sin is_differentiable_in x
sin is_differentiable_in x by SIN_COS:69;
hence (#Z n) * sin is_differentiable_in x by TAYLOR_1:3; :: thesis: verum

end;

then A11: (#Z n) * sin is_differentiable_on Z by A2, FDIFF_1:16;
A12: for x being Real st x in Z holds
(((#Z n) * sin ) `| Z) . x = (n * ((sin . x) #Z (n - 1))) * (cos . x)

proof

let x be Real; :: thesis: ( x in Z implies (((#Z n) * sin ) `| Z) . x = (n * ((sin . x) #Z (n - 1))) * (cos . x) )
assume A13: x in Z ; :: thesis: (((#Z n) * sin ) `| Z) . x = (n * ((sin . x) #Z (n - 1))) * (cos . x)
sin is_differentiable_in x by SIN_COS:69;
then diff ((#Z n) * sin ),x = (n * ((sin . x) #Z (n - 1))) * (diff sin ,x) by TAYLOR_1:3
.= (n * ((sin . x) #Z (n - 1))) * (cos . x) by SIN_COS:69 ;
hence (((#Z n) * sin ) `| Z) . x = (n * ((sin . x) #Z (n - 1))) * (cos . x) by A11, A13, FDIFF_1:def 8; :: thesis: verum

end;

now

let x be Real; :: thesis: ( x in Z implies (((cos * f) (#) ((#Z n) * sin )) `| Z) . x = (n * ((sin . x) #Z (n - 1))) * (cos . ((n + 1) * x)) )
assume A14: x in Z ; :: thesis: (((cos * f) (#) ((#Z n) * sin )) `| Z) . x = (n * ((sin . x) #Z (n - 1))) * (cos . ((n + 1) * x))
A15: n - 1 is Element of NAT by A1, NAT_1:21;
(((cos * f) (#) ((#Z n) * sin )) `| Z) . x = ((((#Z n) * sin ) . x) * (diff (cos * f),x)) + (((cos * f) . x) * (diff ((#Z n) * sin ),x)) by A1, A7, A11, A14, FDIFF_1:29
.= (((#Z n) . (sin . x)) * (diff (cos * f),x)) + (((cos * f) . x) * (diff ((#Z n) * sin ),x)) by A2, A14, FUNCT_1:22
.= (((sin . x) #Z n) * (diff (cos * f),x)) + (((cos * f) . x) * (diff ((#Z n) * sin ),x)) by TAYLOR_1:def 1
.= (((sin . x) #Z n) * (((cos * f) `| Z) . x)) + (((cos * f) . x) * (diff ((#Z n) * sin ),x)) by A7, A14, FDIFF_1:def 8
.= (((sin . x) #Z n) * (- (n * (sin . (n * x))))) + (((cos * f) . x) * (diff ((#Z n) * sin ),x)) by A8, A14
.= (((sin . x) #Z n) * (- (n * (sin . (n * x))))) + ((cos . (f . x)) * (diff ((#Z n) * sin ),x)) by A2, A14, FUNCT_1:22
.= (((sin . x) #Z n) * (- (n * (sin . (n * x))))) + ((cos . (n * x)) * (diff ((#Z n) * sin ),x)) by A1, A14
.= (((sin . x) #Z n) * (- (n * (sin . (n * x))))) + ((cos . (n * x)) * ((((#Z n) * sin ) `| Z) . x)) by A11, A14, FDIFF_1:def 8
.= (((sin . x) #Z n) * (- (n * (sin . (n * x))))) + ((cos . (n * x)) * ((n * ((sin . x) #Z (n - 1))) * (cos . x))) by A12, A14
.= (((sin . x) #Z ((n - 1) + 1)) * (- (n * (sin . (n * x))))) + ((((cos . (n * x)) * n) * ((sin . x) #Z (n - 1))) * (cos . x))
.= ((((sin . x) #Z (n - 1)) * ((sin . x) #Z 1)) * (- (n * (sin . (n * x))))) + ((((cos . (n * x)) * n) * ((sin . x) #Z (n - 1))) * (cos . x)) by A15, TAYLOR_1:1
.= ((((sin . x) #Z (n - 1)) * (sin . x)) * (- (n * (sin . (n * x))))) + ((((cos . (n * x)) * n) * ((sin . x) #Z (n - 1))) * (cos . x)) by PREPOWER:45
.= (n * ((sin . x) #Z (n - 1))) * (((cos . (n * x)) * (cos . x)) - ((sin . x) * (sin . (n * x))))
.= (n * ((sin . x) #Z (n - 1))) * (cos . (x + (n * x))) by SIN_COS:79
.= (n * ((sin . x) #Z (n - 1))) * (cos . ((n + 1) * x)) ;
hence (((cos * f) (#) ((#Z n) * sin )) `| Z) . x = (n * ((sin . x) #Z (n - 1))) * (cos . ((n + 1) * x)) ; :: thesis: verum

end;

hence ( (cos * f) (#) ((#Z n) * sin ) is_differentiable_on Z & ( for x being Real st x in Z holds
(((cos * f) (#) ((#Z n) * sin )) `| Z) . x = (n * ((sin . x) #Z (n - 1))) * (cos . ((n + 1) * x)) ) ) by A1, A7, A11, FDIFF_1:29; :: thesis: verum