let Z be open Subset of REAL ; :: thesis: for f1 being PartFunc of REAL ,REAL st Z c= dom (f1 + (2 (#) cos )) & ( for x being Real st x in Z holds
f1 . x = 1 ) holds
( f1 + (2 (#) cos ) is_differentiable_on Z & ( for x being Real st x in Z holds
((f1 + (2 (#) cos )) `| Z) . x = - (2 * (sin . x)) ) )
let f1 be PartFunc of REAL ,REAL ; :: thesis: ( Z c= dom (f1 + (2 (#) cos )) & ( for x being Real st x in Z holds
f1 . x = 1 ) implies ( f1 + (2 (#) cos ) is_differentiable_on Z & ( for x being Real st x in Z holds
((f1 + (2 (#) cos )) `| Z) . x = - (2 * (sin . x)) ) ) )
assume that
A1: Z c= dom (f1 + (2 (#) cos )) and
A2: for x being Real st x in Z holds
f1 . x = 1 ; :: thesis: ( f1 + (2 (#) cos ) is_differentiable_on Z & ( for x being Real st x in Z holds
((f1 + (2 (#) cos )) `| Z) . x = - (2 * (sin . x)) ) )
A3: Z c= (dom f1) /\ (dom (2 (#) cos )) by A1, VALUED_1:def 1;
then A4: Z c= dom f1 by XBOOLE_1:18;
A5: cos is_differentiable_on Z by FDIFF_1:34, SIN_COS:72;
A6: for x being Real st x in Z holds
f1 . x = (0 * x) + 1 by A2;
then A7: f1 is_differentiable_on Z by A4, FDIFF_1:31;
Z c= dom (2 (#) cos ) by A3, XBOOLE_1:18;
then A8: 2 (#) cos is_differentiable_on Z by A5, FDIFF_1:28;
for x being Real st x in Z holds
((f1 + (2 (#) cos )) `| Z) . x = - (2 * (sin . x))

proof

let x be Real; :: thesis: ( x in Z implies ((f1 + (2 (#) cos )) `| Z) . x = - (2 * (sin . x)) )
A9: cos is_differentiable_in x by SIN_COS:68;
assume A10: x in Z ; :: thesis: ((f1 + (2 (#) cos )) `| Z) . x = - (2 * (sin . x))
then ((f1 + (2 (#) cos )) `| Z) . x = (diff f1,x) + (diff (2 (#) cos ),x) by A1, A7, A8, FDIFF_1:26
.= ((f1 `| Z) . x) + (diff (2 (#) cos ),x) by A7, A10, FDIFF_1:def 8
.= ((f1 `| Z) . x) + (2 * (diff cos ,x)) by A9, FDIFF_1:23
.= 0 + (2 * (diff cos ,x)) by A4, A6, A10, FDIFF_1:31
.= 0 + (2 * (- (sin . x))) by SIN_COS:68
.= - (2 * (sin . x)) ;
hence ((f1 + (2 (#) cos )) `| Z) . x = - (2 * (sin . x)) ; :: thesis: verum

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, A7, A8, FDIFF_1:26; :: thesis: verum