let n be Element of NAT ; :: thesis: 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 ; :: thesis: ( 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 ; :: thesis: ( (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;
then for y being set st y in Z holds
y in dom (cos ^ ) by FUNCT_1:21;
then A5: Z c= dom (cos ^ ) by TARSKI:def 3;
A6: ( cos ^ is_differentiable_on Z & ( for x being Real st x in Z holds
((cos ^ ) `| Z) . x = (sin . x) / ((cos . x) ^2 ) ) ) by A3, FDIFF_4:39;

now

let x be Real; :: thesis: ( x in Z implies (#Z n) * (cos ^ ) is_differentiable_in x )
assume x in Z ; :: thesis: (#Z n) * (cos ^ ) is_differentiable_in x
then cos ^ is_differentiable_in x by A6, FDIFF_1:16;
hence (#Z n) * (cos ^ ) is_differentiable_in x by TAYLOR_1:3; :: thesis: verum

end;

then A7: (#Z n) * (cos ^ ) is_differentiable_on Z by A4, FDIFF_1:16;
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; :: thesis: ( x in Z implies (((1 / n) (#) ((#Z n) * (cos ^ ))) `| Z) . x = (sin . x) / ((cos . x) #Z (n + 1)) )
assume A8: x in Z ; :: thesis: (((1 / n) (#) ((#Z n) * (cos ^ ))) `| Z) . x = (sin . x) / ((cos . x) #Z (n + 1))
then A9: cos ^ is_differentiable_in x by A6, FDIFF_1:16;
A10: (cos ^ ) . x = (cos . x) " by A5, A8, RFUNCT_1:def 8
.= 1 / (cos . x) by XCMPLX_1:217 ;
(((1 / n) (#) ((#Z n) * (cos ^ ))) `| Z) . x = (1 / n) * (diff ((#Z n) * (cos ^ )),x) by A1, A7, A8, FDIFF_1:28
.= (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 A6, A8, FDIFF_1:def 8
.= (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_1:107
.= ((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:52
.= ((sin . x) / ((cos . x) #Z 2)) / ((cos . x) #Z (n - 1)) by XCMPLX_1:100
.= (sin . x) / (((cos . x) #Z 2) * ((cos . x) #Z (n - 1))) by XCMPLX_1:79
.= (sin . x) / ((cos . x) #Z (2 + (n - 1))) by A3, A8, PREPOWER:54
.= (sin . x) / ((cos . x) #Z (n + 1)) ;
hence (((1 / n) (#) ((#Z n) * (cos ^ ))) `| Z) . x = (sin . x) / ((cos . x) #Z (n + 1)) ; :: thesis: 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, A7, FDIFF_1:28; :: thesis: verum