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

proof

let x be Real; :: thesis: ( x in Z implies cos * arctan is_differentiable_in x )
assume A3: x in Z ; :: thesis: cos * arctan is_differentiable_in x
arctan is_differentiable_on Z by A2, SIN_COS9:81;
then A4: arctan is_differentiable_in x by A3, FDIFF_1:16;
cos is_differentiable_in arctan . x by SIN_COS:68;
hence cos * arctan is_differentiable_in x by A4, FDIFF_2:13; :: thesis: verum

end;

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

proof

let x be Real; :: thesis: ( x in Z implies ((cos * arctan ) `| Z) . x = - ((sin . (arctan . x)) / (1 + (x ^2 ))) )
assume A6: x in Z ; :: thesis: ((cos * arctan ) `| Z) . x = - ((sin . (arctan . x)) / (1 + (x ^2 )))
A7: arctan is_differentiable_on Z by A2, SIN_COS9:81;
then A8: arctan is_differentiable_in x by A6, FDIFF_1:16;
A9: cos is_differentiable_in arctan . x by SIN_COS:68;
((cos * arctan ) `| Z) . x = diff (cos * arctan ),x by A5, A6, FDIFF_1:def 8
.= (diff cos ,(arctan . x)) * (diff arctan ,x) by A8, A9, FDIFF_2:13
.= (- (sin . (arctan . x))) * (diff arctan ,x) by SIN_COS:68
.= - ((sin . (arctan . x)) * (diff arctan ,x))
.= - ((sin . (arctan . x)) * ((arctan `| Z) . x)) by A6, A7, FDIFF_1:def 8
.= - ((sin . (arctan . x)) * (1 / (1 + (x ^2 )))) by A2, A6, SIN_COS9:81
.= - ((sin . (arctan . x)) / (1 + (x ^2 ))) ;
hence ((cos * arctan ) `| Z) . x = - ((sin . (arctan . x)) / (1 + (x ^2 ))) ; :: thesis: verum

end;

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