let Z be open Subset of REAL; 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; ( 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
; ( 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_1:26, SIN_COS:67;
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_1:23;
Z c= dom (2 (#) cos)
by A3, XBOOLE_1:18;
then A8:
2 (#) cos is_differentiable_on Z
by A5, FDIFF_1:20;
for x being Real st x in Z holds
((f1 + (2 (#) cos)) `| Z) . x = - (2 * (sin . x))
proof
let x be
Real;
( x in Z implies ((f1 + (2 (#) cos)) `| Z) . x = - (2 * (sin . x)) )
A9:
cos is_differentiable_in x
by SIN_COS:63;
assume A10:
x in Z
;
((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_1:18
.=
((f1 `| Z) . x) + (diff ((2 (#) cos),x))
by A7, A10, FDIFF_1:def 7
.=
((f1 `| Z) . x) + (2 * (diff (cos,x)))
by A9, FDIFF_1:15
.=
0 + (2 * (diff (cos,x)))
by A4, A6, A10, FDIFF_1:23
.=
0 + (2 * (- (sin . x)))
by SIN_COS:63
.=
- (2 * (sin . x))
;
hence
((f1 + (2 (#) cos)) `| Z) . x = - (2 * (sin . x))
;
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_1:18; verum