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_1:27
.= (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_1:27; :: thesis: verum