let Z be open Subset of REAL ; :: thesis:
let f1 be PartFunc of REAL ,REAL ; :: thesis:
assume that
A1: Z c= dom (f1 + (2 (#) cos )) and
A2: for x being Real st x in Z holds
f1 . x = 1 ; :: thesis:
Z c= (dom f1) /\ (dom (2 (#) cos )) by A1, VALUED_1:def 1;
then A3: ( Z c= dom f1 & Z c= dom (2 (#) cos ) ) by XBOOLE_1:18;
A4: for x being Real st x in Z holds
f1 . x = (0 * x) + 1 by A2;
then A5: ( f1 is_differentiable_on Z & ( for x being Real st x in Z holds
(f1 `| Z) . x = 0 ) ) by A3, FDIFF_1:31;
cos is_differentiable_on Z by FDIFF_1:34, SIN_COS:72;
then A6: 2 (#) cos is_differentiable_on Z by A3, FDIFF_1:28;
for x being Real st x in Z holds
((f1 + (2 (#) cos )) `| Z) . x = - (2 * (sin . x))

let x be Real; :: thesis:
assume A7: x in Z ; :: thesis:
A8: cos is_differentiable_in x by SIN_COS:68;
((f1 + (2 (#) cos )) `| Z) . x = (diff f1,x) + (diff (2 (#) cos ),x) by A1, A5, A6, A7, FDIFF_1:26
.= ((f1 `| Z) . x) + (diff (2 (#) cos ),x) by A5, A7, FDIFF_1:def 8
.= ((f1 `| Z) . x) + (2 * (diff cos ,x)) by A8, FDIFF_1:23
.= 0 + (2 * (diff cos ,x)) by A3, A4, A7, FDIFF_1:31
.= 0 + (2 * (- (sin . x))) by SIN_COS:68
.= - (2 * (sin . x)) ;
hence ((f1 + (2 (#) cos )) `| Z) . x = - (2 * (sin . x)) ; :: thesis:

end;

hence ( f1 + (2 (#) cos ) is_differentiable_on Z & ( for x being Real st x in Z holds
((f1 + (2 (#) cos )) `| Z) . x = - (2 * (sin . x)) ) ) by A1, A5, A6, FDIFF_1:26; :: thesis: