let Z be open Subset of REAL ; :: thesis: ( Z c= dom (2 (#) ((#R (1 / 2)) * sin )) & ( for x being Real st x in Z holds
sin . x > 0 ) implies ( 2 (#) ((#R (1 / 2)) * sin ) is_differentiable_on Z & ( for x being Real st x in Z holds
((2 (#) ((#R (1 / 2)) * sin )) `| Z) . x = (cos . x) * ((sin . x) #R (- (1 / 2))) ) ) )
assume that
A1: Z c= dom (2 (#) ((#R (1 / 2)) * sin )) and
A2: for x being Real st x in Z holds
sin . x > 0 ; :: thesis: ( 2 (#) ((#R (1 / 2)) * sin ) is_differentiable_on Z & ( for x being Real st x in Z holds
((2 (#) ((#R (1 / 2)) * sin )) `| Z) . x = (cos . x) * ((sin . x) #R (- (1 / 2))) ) )

A3: now

let x be Real; :: thesis: ( x in Z implies (#R (1 / 2)) * sin is_differentiable_in x )
assume x in Z ; :: thesis: (#R (1 / 2)) * sin is_differentiable_in x
then ( sin is_differentiable_in x & sin . x > 0 ) by A2, SIN_COS:69;
hence (#R (1 / 2)) * sin is_differentiable_in x by TAYLOR_1:22; :: thesis: verum

end;

Z c= dom ((#R (1 / 2)) * sin ) by A1, VALUED_1:def 5;
then A4: (#R (1 / 2)) * sin is_differentiable_on Z by A3, FDIFF_1:16;
for x being Real st x in Z holds
((2 (#) ((#R (1 / 2)) * sin )) `| Z) . x = (cos . x) * ((sin . x) #R (- (1 / 2)))

proof

let x be Real; :: thesis: ( x in Z implies ((2 (#) ((#R (1 / 2)) * sin )) `| Z) . x = (cos . x) * ((sin . x) #R (- (1 / 2))) )
assume A5: x in Z ; :: thesis: ((2 (#) ((#R (1 / 2)) * sin )) `| Z) . x = (cos . x) * ((sin . x) #R (- (1 / 2)))
then A6: ( sin is_differentiable_in x & sin . x > 0 ) by A2, SIN_COS:69;
((2 (#) ((#R (1 / 2)) * sin )) `| Z) . x = 2 * (diff ((#R (1 / 2)) * sin ),x) by A1, A4, A5, FDIFF_1:28
.= 2 * (((1 / 2) * ((sin . x) #R ((1 / 2) - 1))) * (diff sin ,x)) by A6, TAYLOR_1:22
.= 2 * (((1 / 2) * ((sin . x) #R ((1 / 2) - 1))) * (cos . x)) by SIN_COS:69
.= (cos . x) * ((sin . x) #R (- (1 / 2))) ;
hence ((2 (#) ((#R (1 / 2)) * sin )) `| Z) . x = (cos . x) * ((sin . x) #R (- (1 / 2))) ; :: thesis: verum

end;

hence ( 2 (#) ((#R (1 / 2)) * sin ) is_differentiable_on Z & ( for x being Real st x in Z holds
((2 (#) ((#R (1 / 2)) * sin )) `| Z) . x = (cos . x) * ((sin . x) #R (- (1 / 2))) ) ) by A1, A4, FDIFF_1:28; :: thesis: verum