let Z be open Subset of REAL; ( Z c= dom (sin + (#R (1 / 2))) implies ( sin + (#R (1 / 2)) is_differentiable_on Z & ( for x being Real st x in Z holds
((sin + (#R (1 / 2))) `| Z) . x = (cos . x) + ((1 / 2) * (x #R (- (1 / 2)))) ) ) )
assume A1:
Z c= dom (sin + (#R (1 / 2)))
; ( sin + (#R (1 / 2)) is_differentiable_on Z & ( for x being Real st x in Z holds
((sin + (#R (1 / 2))) `| Z) . x = (cos . x) + ((1 / 2) * (x #R (- (1 / 2)))) ) )
then
Z c= (dom (#R (1 / 2))) /\ (dom sin)
by VALUED_1:def 1;
then A2:
Z c= dom (#R (1 / 2))
by XBOOLE_1:18;
then A3:
#R (1 / 2) is_differentiable_on Z
by Lm3;
A4:
sin is_differentiable_on Z
by FDIFF_1:26, SIN_COS:68;
now for x being Real st x in Z holds
((sin + (#R (1 / 2))) `| Z) . x = (cos . x) + ((1 / 2) * (x #R (- (1 / 2))))let x be
Real;
( x in Z implies ((sin + (#R (1 / 2))) `| Z) . x = (cos . x) + ((1 / 2) * (x #R (- (1 / 2)))) )assume A5:
x in Z
;
((sin + (#R (1 / 2))) `| Z) . x = (cos . x) + ((1 / 2) * (x #R (- (1 / 2))))then ((sin + (#R (1 / 2))) `| Z) . x =
(diff (sin,x)) + (diff ((#R (1 / 2)),x))
by A1, A3, A4, FDIFF_1:18
.=
(cos . x) + (diff ((#R (1 / 2)),x))
by SIN_COS:64
.=
(cos . x) + (((#R (1 / 2)) `| Z) . x)
by A3, A5, FDIFF_1:def 7
.=
(cos . x) + ((1 / 2) * (x #R (- (1 / 2))))
by A2, A5, Lm3
;
hence
((sin + (#R (1 / 2))) `| Z) . x = (cos . x) + ((1 / 2) * (x #R (- (1 / 2))))
;
verum end;
hence
( sin + (#R (1 / 2)) is_differentiable_on Z & ( for x being Real st x in Z holds
((sin + (#R (1 / 2))) `| Z) . x = (cos . x) + ((1 / 2) * (x #R (- (1 / 2)))) ) )
by A1, A3, A4, FDIFF_1:18; verum