let Z be open Subset of REAL; :: thesis: ( Z c= dom (sin (#) arctan) & Z c= ].(- 1),1.[ implies ( sin (#) arctan is_differentiable_on Z & ( for x being Real st x in Z holds
((sin (#) arctan) `| Z) . x = ((cos . x) * (arctan . x)) + ((sin . x) / (1 + (x ^2))) ) ) )
assume that
A1: Z c= dom (sin (#) arctan) and
A2: Z c= ].(- 1),1.[ ; :: thesis: ( sin (#) arctan is_differentiable_on Z & ( for x being Real st x in Z holds
((sin (#) arctan) `| Z) . x = ((cos . x) * (arctan . x)) + ((sin . x) / (1 + (x ^2))) ) )
A3: arctan is_differentiable_on Z by A2, SIN_COS9:81;
A4: for x being Real st x in Z holds
sin is_differentiable_in x by SIN_COS:64;
Z c= (dom sin) /\ (dom arctan) by A1, VALUED_1:def 4;
then Z c= dom sin by XBOOLE_1:18;
then A5: sin is_differentiable_on Z by A4, FDIFF_1:9;
for x being Real st x in Z holds
((sin (#) arctan) `| Z) . x = ((cos . x) * (arctan . x)) + ((sin . x) / (1 + (x ^2)))

proof

let x be Real; :: thesis: ( x in Z implies ((sin (#) arctan) `| Z) . x = ((cos . x) * (arctan . x)) + ((sin . x) / (1 + (x ^2))) )
assume A6: x in Z ; :: thesis: ((sin (#) arctan) `| Z) . x = ((cos . x) * (arctan . x)) + ((sin . x) / (1 + (x ^2)))
then ((sin (#) arctan) `| Z) . x = ((arctan . x) * (diff (sin,x))) + ((sin . x) * (diff (arctan,x))) by A1, A5, A3, FDIFF_1:21
.= ((arctan . x) * (cos . x)) + ((sin . x) * (diff (arctan,x))) by SIN_COS:64
.= ((cos . x) * (arctan . x)) + ((sin . x) * ((arctan `| Z) . x)) by A3, A6, FDIFF_1:def 7
.= ((cos . x) * (arctan . x)) + ((sin . x) * (1 / (1 + (x ^2)))) by A2, A6, SIN_COS9:81
.= ((cos . x) * (arctan . x)) + ((sin . x) / (1 + (x ^2))) ;
hence ((sin (#) arctan) `| Z) . x = ((cos . x) * (arctan . x)) + ((sin . x) / (1 + (x ^2))) ; :: thesis: verum

end;

hence ( sin (#) arctan is_differentiable_on Z & ( for x being Real st x in Z holds
((sin (#) arctan) `| Z) . x = ((cos . x) * (arctan . x)) + ((sin . x) / (1 + (x ^2))) ) ) by A1, A5, A3, FDIFF_1:21; :: thesis: verum