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

proof

let x be Real; :: thesis: ( x in Z implies arctan * sin is_differentiable_in x )
assume A3: x in Z ; :: thesis: arctan * sin is_differentiable_in x
A4: sin is_differentiable_in x by SIN_COS:69;
( sin . x > - 1 & sin . x < 1 ) by A2, A3;
hence arctan * sin is_differentiable_in x by A4, SIN_COS9:85; :: thesis: verum

end;

then A5: arctan * sin is_differentiable_on Z by A1, FDIFF_1:16;
for x being Real st x in Z holds
((arctan * sin ) `| Z) . x = (cos . x) / (1 + ((sin . x) ^2 ))

proof

let x be Real; :: thesis: ( x in Z implies ((arctan * sin ) `| Z) . x = (cos . x) / (1 + ((sin . x) ^2 )) )
assume A6: x in Z ; :: thesis: ((arctan * sin ) `| Z) . x = (cos . x) / (1 + ((sin . x) ^2 ))
A7: sin is_differentiable_in x by SIN_COS:69;
A8: ( sin . x > - 1 & sin . x < 1 ) by A2, A6;
((arctan * sin ) `| Z) . x = diff (arctan * sin ),x by A5, A6, FDIFF_1:def 8
.= (diff sin ,x) / (1 + ((sin . x) ^2 )) by A7, A8, SIN_COS9:85
.= (cos . x) / (1 + ((sin . x) ^2 )) by SIN_COS:69 ;
hence ((arctan * sin ) `| Z) . x = (cos . x) / (1 + ((sin . x) ^2 )) ; :: thesis: verum

end;

hence ( arctan * sin is_differentiable_on Z & ( for x being Real st x in Z holds
((arctan * sin ) `| Z) . x = (cos . x) / (1 + ((sin . x) ^2 )) ) ) by A1, AA, FDIFF_1:16; :: thesis: verum