let a be Real; :: thesis: for Z being open Subset of REAL
for f being PartFunc of REAL ,REAL st Z c= dom ((id Z) (#) (arccos * f)) & ( for x being Real st x in Z holds
( f . x = x / a & f . x > - 1 & f . x < 1 ) ) holds
( (id Z) (#) (arccos * f) is_differentiable_on Z & ( for x being Real st x in Z holds
(((id Z) (#) (arccos * f)) `| Z) . x = (arccos . (x / a)) - (x / (a * (sqrt (1 - ((x / a) ^2 ))))) ) )
let Z be open Subset of REAL ; :: thesis: for f being PartFunc of REAL ,REAL st Z c= dom ((id Z) (#) (arccos * f)) & ( for x being Real st x in Z holds
( f . x = x / a & f . x > - 1 & f . x < 1 ) ) holds
( (id Z) (#) (arccos * f) is_differentiable_on Z & ( for x being Real st x in Z holds
(((id Z) (#) (arccos * f)) `| Z) . x = (arccos . (x / a)) - (x / (a * (sqrt (1 - ((x / a) ^2 ))))) ) )
let f be PartFunc of REAL ,REAL ; :: thesis: ( Z c= dom ((id Z) (#) (arccos * f)) & ( for x being Real st x in Z holds
( f . x = x / a & f . x > - 1 & f . x < 1 ) ) implies ( (id Z) (#) (arccos * f) is_differentiable_on Z & ( for x being Real st x in Z holds
(((id Z) (#) (arccos * f)) `| Z) . x = (arccos . (x / a)) - (x / (a * (sqrt (1 - ((x / a) ^2 ))))) ) ) )
assume that
A1:
Z c= dom ((id Z) (#) (arccos * f))
and
A2:
for x being Real st x in Z holds
( f . x = x / a & f . x > - 1 & f . x < 1 )
; :: thesis: ( (id Z) (#) (arccos * f) is_differentiable_on Z & ( for x being Real st x in Z holds
(((id Z) (#) (arccos * f)) `| Z) . x = (arccos . (x / a)) - (x / (a * (sqrt (1 - ((x / a) ^2 ))))) ) )
for x being Real st x in Z holds
f . x = ((1 / a) * x) + 0
then A3:
for x being Real st x in Z holds
( f . x = ((1 / a) * x) + 0 & f . x > - 1 & f . x < 1 )
by A2;
A4:
Z c= (dom (id Z)) /\ (dom (arccos * f))
by A1, VALUED_1:def 4;
then A5:
Z c= dom (id Z)
by XBOOLE_1:18;
A6:
Z c= dom (arccos * f)
by A4, XBOOLE_1:18;
A7:
for x being Real st x in Z holds
(id Z) . x = (1 * x) + 0
by FUNCT_1:35;
then A8:
( id Z is_differentiable_on Z & ( for x being Real st x in Z holds
((id Z) `| Z) . x = 1 ) )
by A5, FDIFF_1:31;
A9:
( arccos * f is_differentiable_on Z & ( for x being Real st x in Z holds
((arccos * f) `| Z) . x = - ((1 / a) / (sqrt (1 - ((((1 / a) * x) + 0 ) ^2 )))) ) )
by A3, A6, Th15;
A10:
for x being Real st x in Z holds
((arccos * f) `| Z) . x = - (1 / (a * (sqrt (1 - ((x / a) ^2 )))))
for x being Real st x in Z holds
(((id Z) (#) (arccos * f)) `| Z) . x = (arccos . (x / a)) - (x / (a * (sqrt (1 - ((x / a) ^2 )))))
proof
let x be
Real;
:: thesis: ( x in Z implies (((id Z) (#) (arccos * f)) `| Z) . x = (arccos . (x / a)) - (x / (a * (sqrt (1 - ((x / a) ^2 ))))) )
assume A11:
x in Z
;
:: thesis: (((id Z) (#) (arccos * f)) `| Z) . x = (arccos . (x / a)) - (x / (a * (sqrt (1 - ((x / a) ^2 )))))
then A12:
(arccos * f) . x =
arccos . (f . x)
by A6, FUNCT_1:22
.=
arccos . (x / a)
by A2, A11
;
(((id Z) (#) (arccos * f)) `| Z) . x =
(((arccos * f) . x) * (diff (id Z),x)) + (((id Z) . x) * (diff (arccos * f),x))
by A1, A8, A9, A11, FDIFF_1:29
.=
(((arccos * f) . x) * (((id Z) `| Z) . x)) + (((id Z) . x) * (diff (arccos * f),x))
by A8, A11, FDIFF_1:def 8
.=
(((arccos * f) . x) * 1) + (((id Z) . x) * (diff (arccos * f),x))
by A5, A7, A11, FDIFF_1:31
.=
(((arccos * f) . x) * 1) + (((id Z) . x) * (((arccos * f) `| Z) . x))
by A9, A11, FDIFF_1:def 8
.=
((arccos * f) . x) + (x * (((arccos * f) `| Z) . x))
by A11, FUNCT_1:35
.=
(arccos . (x / a)) + (x * (- (1 / (a * (sqrt (1 - ((x / a) ^2 )))))))
by A10, A11, A12
.=
(arccos . (x / a)) - (x * (1 / (a * (sqrt (1 - ((x / a) ^2 ))))))
.=
(arccos . (x / a)) - (x / (a * (sqrt (1 - ((x / a) ^2 )))))
by XCMPLX_1:100
;
hence
(((id Z) (#) (arccos * f)) `| Z) . x = (arccos . (x / a)) - (x / (a * (sqrt (1 - ((x / a) ^2 )))))
;
:: thesis: verum
end;
hence
( (id Z) (#) (arccos * f) is_differentiable_on Z & ( for x being Real st x in Z holds
(((id Z) (#) (arccos * f)) `| Z) . x = (arccos . (x / a)) - (x / (a * (sqrt (1 - ((x / a) ^2 ))))) ) )
by A1, A8, A9, FDIFF_1:29; :: thesis: verum