let Z be open Subset of REAL; :: thesis: ( Z c= dom (sin + cos) implies ( sin + cos is_differentiable_on Z & ( for x being Real st x in Z holds
((sin + cos) `| Z) . x = (cos . x) - (sin . x) ) ) )
A1: ( sin is_differentiable_on Z & cos is_differentiable_on Z ) by FDIFF_1:34, SIN_COS:72, SIN_COS:73;
assume A2: Z c= dom (sin + cos) ; :: thesis: ( sin + cos is_differentiable_on Z & ( for x being Real st x in Z holds
((sin + cos) `| Z) . x = (cos . x) - (sin . x) ) )

now

let x be Real; :: thesis: ( x in Z implies ((sin + cos) `| Z) . x = (cos . x) - (sin . x) )
assume x in Z ; :: thesis: ((sin + cos) `| Z) . x = (cos . x) - (sin . x)
hence ((sin + cos) `| Z) . x = (diff (sin,x)) + (diff (cos,x)) by A2, A1, FDIFF_1:26
.= (cos . x) + (diff (cos,x)) by SIN_COS:69
.= (cos . x) + (- (sin . x)) by SIN_COS:68
.= (cos . x) - (sin . x) ;
:: thesis: verum

end;

hence ( sin + cos is_differentiable_on Z & ( for x being Real st x in Z holds
((sin + cos) `| Z) . x = (cos . x) - (sin . x) ) ) by A2, A1, FDIFF_1:26; :: thesis: verum