let Z be open Subset of REAL ; :: thesis: ( Z c= dom (arctan * cos ) & ( for x being Real st x in Z holds
( cos . x > - 1 & cos . x < 1 ) ) implies ( arctan * cos is_differentiable_on Z & ( for x being Real st x in Z holds
((arctan * cos ) `| Z) . x = - ((sin . x) / (1 + ((cos . x) ^2 ))) ) ) )
assume that
A1: Z c= dom (arctan * cos ) and
A2: for x being Real st x in Z holds
( cos . x > - 1 & cos . x < 1 ) ; :: thesis: ( arctan * cos is_differentiable_on Z & ( for x being Real st x in Z holds
((arctan * cos ) `| Z) . x = - ((sin . x) / (1 + ((cos . x) ^2 ))) ) )
A3: for x being Real st x in Z holds
arctan * cos is_differentiable_in x

proof

let x be Real; :: thesis: ( x in Z implies arctan * cos is_differentiable_in x )
assume x in Z ; :: thesis: arctan * cos is_differentiable_in x
then A4: ( cos . x > - 1 & cos . x < 1 ) by A2;
cos is_differentiable_in x by SIN_COS:68;
hence arctan * cos is_differentiable_in x by A4, SIN_COS9:85; :: thesis: verum

end;

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

proof

let x be Real; :: thesis: ( x in Z implies ((arctan * cos ) `| Z) . x = - ((sin . x) / (1 + ((cos . x) ^2 ))) )
A6: cos is_differentiable_in x by SIN_COS:68;
assume A7: x in Z ; :: thesis: ((arctan * cos ) `| Z) . x = - ((sin . x) / (1 + ((cos . x) ^2 )))
then A8: ( cos . x > - 1 & cos . x < 1 ) by A2;
((arctan * cos ) `| Z) . x = diff (arctan * cos ),x by A5, A7, FDIFF_1:def 8
.= (diff cos ,x) / (1 + ((cos . x) ^2 )) by A6, A8, SIN_COS9:85
.= (- (sin . x)) / (1 + ((cos . x) ^2 )) by SIN_COS:68
.= - ((sin . x) / (1 + ((cos . x) ^2 ))) ;
hence ((arctan * cos ) `| Z) . x = - ((sin . x) / (1 + ((cos . x) ^2 ))) ; :: thesis: verum

end;

hence ( arctan * cos is_differentiable_on Z & ( for x being Real st x in Z holds
((arctan * cos ) `| Z) . x = - ((sin . x) / (1 + ((cos . x) ^2 ))) ) ) by A1, A3, FDIFF_1:16; :: thesis: verum