let a be Real; for Z being open Subset of REAL
for f, f1 being PartFunc of REAL,REAL st Z c= dom (f1 - ((1 / (4 * a)) (#) (sin * f))) & ( for x being Real st x in Z holds
( f1 . x = x / 2 & f . x = (2 * a) * x ) ) & a <> 0 holds
( f1 - ((1 / (4 * a)) (#) (sin * f)) is_differentiable_on Z & ( for x being Real st x in Z holds
((f1 - ((1 / (4 * a)) (#) (sin * f))) `| Z) . x = (sin (a * x)) ^2 ) )
let Z be open Subset of REAL; for f, f1 being PartFunc of REAL,REAL st Z c= dom (f1 - ((1 / (4 * a)) (#) (sin * f))) & ( for x being Real st x in Z holds
( f1 . x = x / 2 & f . x = (2 * a) * x ) ) & a <> 0 holds
( f1 - ((1 / (4 * a)) (#) (sin * f)) is_differentiable_on Z & ( for x being Real st x in Z holds
((f1 - ((1 / (4 * a)) (#) (sin * f))) `| Z) . x = (sin (a * x)) ^2 ) )
let f, f1 be PartFunc of REAL,REAL; ( Z c= dom (f1 - ((1 / (4 * a)) (#) (sin * f))) & ( for x being Real st x in Z holds
( f1 . x = x / 2 & f . x = (2 * a) * x ) ) & a <> 0 implies ( f1 - ((1 / (4 * a)) (#) (sin * f)) is_differentiable_on Z & ( for x being Real st x in Z holds
((f1 - ((1 / (4 * a)) (#) (sin * f))) `| Z) . x = (sin (a * x)) ^2 ) ) )
assume that
A1:
Z c= dom (f1 - ((1 / (4 * a)) (#) (sin * f)))
and
A2:
for x being Real st x in Z holds
( f1 . x = x / 2 & f . x = (2 * a) * x )
and
A3:
a <> 0
; ( f1 - ((1 / (4 * a)) (#) (sin * f)) is_differentiable_on Z & ( for x being Real st x in Z holds
((f1 - ((1 / (4 * a)) (#) (sin * f))) `| Z) . x = (sin (a * x)) ^2 ) )
A4:
Z c= (dom f1) /\ (dom ((1 / (4 * a)) (#) (sin * f)))
by A1, VALUED_1:12;
then A5:
Z c= dom ((1 / (4 * a)) (#) (sin * f))
by XBOOLE_1:18;
A6:
for x being Real st x in Z holds
f1 . x = ((1 / 2) * x) + 0
proof
let x be
Real;
( x in Z implies f1 . x = ((1 / 2) * x) + 0 )
assume
x in Z
;
f1 . x = ((1 / 2) * x) + 0
then f1 . x =
x / 2
by A2
.=
((1 / 2) * x) + 0
;
hence
f1 . x = ((1 / 2) * x) + 0
;
verum
end;
A7:
for x being Real st x in Z holds
f . x = (2 * a) * x
by A2;
then A8:
(1 / (4 * a)) (#) (sin * f) is_differentiable_on Z
by A3, A5, Th43;
A9:
Z c= dom f1
by A4, XBOOLE_1:18;
then A10:
f1 is_differentiable_on Z
by A6, FDIFF_1:23;
for x being Real st x in Z holds
((f1 - ((1 / (4 * a)) (#) (sin * f))) `| Z) . x = (sin (a * x)) ^2
proof
let x be
Real;
( x in Z implies ((f1 - ((1 / (4 * a)) (#) (sin * f))) `| Z) . x = (sin (a * x)) ^2 )
assume A11:
x in Z
;
((f1 - ((1 / (4 * a)) (#) (sin * f))) `| Z) . x = (sin (a * x)) ^2
then ((f1 - ((1 / (4 * a)) (#) (sin * f))) `| Z) . x =
(diff (f1,x)) - (diff (((1 / (4 * a)) (#) (sin * f)),x))
by A1, A8, A10, FDIFF_1:19
.=
((f1 `| Z) . x) - (diff (((1 / (4 * a)) (#) (sin * f)),x))
by A10, A11, FDIFF_1:def 7
.=
((f1 `| Z) . x) - ((((1 / (4 * a)) (#) (sin * f)) `| Z) . x)
by A8, A11, FDIFF_1:def 7
.=
((f1 `| Z) . x) - ((1 / 2) * (cos ((2 * a) * x)))
by A3, A7, A5, A11, Th43
.=
(1 / 2) - ((1 / 2) * (cos ((2 * a) * x)))
by A9, A6, A11, FDIFF_1:23
.=
(1 / 2) * (1 - (cos (2 * (a * x))))
.=
(1 / 2) * (2 * ((sin (a * x)) ^2))
by Lm1
.=
(sin (a * x)) ^2
;
hence
((f1 - ((1 / (4 * a)) (#) (sin * f))) `| Z) . x = (sin (a * x)) ^2
;
verum
end;
hence
( f1 - ((1 / (4 * a)) (#) (sin * f)) is_differentiable_on Z & ( for x being Real st x in Z holds
((f1 - ((1 / (4 * a)) (#) (sin * f))) `| Z) . x = (sin (a * x)) ^2 ) )
by A1, A8, A10, FDIFF_1:19; verum