let Z be open Subset of REAL; for f, f1, f2 being PartFunc of REAL,REAL st Z c= dom (((id Z) (#) arcsin) + ((#R (1 / 2)) * f)) & Z c= ].(- 1),1.[ & f = f1 - f2 & f2 = #Z 2 & ( for x being Real st x in Z holds
( f1 . x = 1 & f . x > 0 & x <> 0 ) ) holds
( ((id Z) (#) arcsin) + ((#R (1 / 2)) * f) is_differentiable_on Z & ( for x being Real st x in Z holds
((((id Z) (#) arcsin) + ((#R (1 / 2)) * f)) `| Z) . x = arcsin . x ) )
let f, f1, f2 be PartFunc of REAL,REAL; ( Z c= dom (((id Z) (#) arcsin) + ((#R (1 / 2)) * f)) & Z c= ].(- 1),1.[ & f = f1 - f2 & f2 = #Z 2 & ( for x being Real st x in Z holds
( f1 . x = 1 & f . x > 0 & x <> 0 ) ) implies ( ((id Z) (#) arcsin) + ((#R (1 / 2)) * f) is_differentiable_on Z & ( for x being Real st x in Z holds
((((id Z) (#) arcsin) + ((#R (1 / 2)) * f)) `| Z) . x = arcsin . x ) ) )
assume that
A1:
Z c= dom (((id Z) (#) arcsin) + ((#R (1 / 2)) * f))
and
A2:
Z c= ].(- 1),1.[
and
A3:
f = f1 - f2
and
A4:
f2 = #Z 2
and
A5:
for x being Real st x in Z holds
( f1 . x = 1 & f . x > 0 & x <> 0 )
; ( ((id Z) (#) arcsin) + ((#R (1 / 2)) * f) is_differentiable_on Z & ( for x being Real st x in Z holds
((((id Z) (#) arcsin) + ((#R (1 / 2)) * f)) `| Z) . x = arcsin . x ) )
A6:
Z c= (dom ((id Z) (#) arcsin)) /\ (dom ((#R (1 / 2)) * f))
by A1, VALUED_1:def 1;
then A7:
Z c= dom ((#R (1 / 2)) * f)
by XBOOLE_1:18;
A8:
Z c= dom ((id Z) (#) arcsin)
by A6, XBOOLE_1:18;
then A9:
(id Z) (#) arcsin is_differentiable_on Z
by A2, Th16;
A10:
for x being Real st x in Z holds
( f1 . x = 1 & f . x > 0 )
by A5;
then A11:
(#R (1 / 2)) * f is_differentiable_on Z
by A3, A4, A7, Th22;
A12:
for x being Real st x in Z holds
x * ((1 - (x #Z 2)) #R (- (1 / 2))) = x / (sqrt (1 - (x ^2)))
for x being Real st x in Z holds
((((id Z) (#) arcsin) + ((#R (1 / 2)) * f)) `| Z) . x = arcsin . x
proof
let x be
Real;
( x in Z implies ((((id Z) (#) arcsin) + ((#R (1 / 2)) * f)) `| Z) . x = arcsin . x )
assume A16:
x in Z
;
((((id Z) (#) arcsin) + ((#R (1 / 2)) * f)) `| Z) . x = arcsin . x
hence ((((id Z) (#) arcsin) + ((#R (1 / 2)) * f)) `| Z) . x =
(diff (((id Z) (#) arcsin),x)) + (diff (((#R (1 / 2)) * f),x))
by A1, A9, A11, FDIFF_1:18
.=
((((id Z) (#) arcsin) `| Z) . x) + (diff (((#R (1 / 2)) * f),x))
by A9, A16, FDIFF_1:def 7
.=
((((id Z) (#) arcsin) `| Z) . x) + ((((#R (1 / 2)) * f) `| Z) . x)
by A11, A16, FDIFF_1:def 7
.=
((arcsin . x) + (x / (sqrt (1 - (x ^2))))) + ((((#R (1 / 2)) * f) `| Z) . x)
by A2, A8, A16, Th16
.=
((arcsin . x) + (x / (sqrt (1 - (x ^2))))) + (- (x * ((1 - (x #Z 2)) #R (- (1 / 2)))))
by A3, A4, A10, A7, A16, Th22
.=
((arcsin . x) + (x / (sqrt (1 - (x ^2))))) - (x * ((1 - (x #Z 2)) #R (- (1 / 2))))
.=
((arcsin . x) + (x / (sqrt (1 - (x ^2))))) - (x / (sqrt (1 - (x ^2))))
by A12, A16
.=
arcsin . x
;
verum
end;
hence
( ((id Z) (#) arcsin) + ((#R (1 / 2)) * f) is_differentiable_on Z & ( for x being Real st x in Z holds
((((id Z) (#) arcsin) + ((#R (1 / 2)) * f)) `| Z) . x = arcsin . x ) )
by A1, A9, A11, FDIFF_1:18; verum