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) ) ) )
assume A1: 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) ) )
A2: sin is_differentiable_on Z by FDIFF_1:34, SIN_COS:73;
A3: cos is_differentiable_on Z by FDIFF_1:34, SIN_COS:72;

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 A1, A2, A3, 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 A1, A2, A3, FDIFF_1:26; :: thesis: verum