let Z be open Subset of REAL; :: thesis: for f being PartFunc of REAL,REAL st Z c= dom ((1 / 2) (#) (arccos * f)) & ( for x being Real st x in Z holds
( f . x = 2 * x & f . x > - 1 & f . x < 1 ) ) holds
( (1 / 2) (#) (arccos * f) is_differentiable_on Z & ( for x being Real st x in Z holds
(((1 / 2) (#) (arccos * f)) `| Z) . x = - (1 / (sqrt (1 - ((2 * x) ^2)))) ) )
let f be PartFunc of REAL,REAL; :: thesis: ( Z c= dom ((1 / 2) (#) (arccos * f)) & ( for x being Real st x in Z holds
( f . x = 2 * x & f . x > - 1 & f . x < 1 ) ) implies ( (1 / 2) (#) (arccos * f) is_differentiable_on Z & ( for x being Real st x in Z holds
(((1 / 2) (#) (arccos * f)) `| Z) . x = - (1 / (sqrt (1 - ((2 * x) ^2)))) ) ) )
assume that
A1: Z c= dom ((1 / 2) (#) (arccos * f)) and
A2: for x being Real st x in Z holds
( f . x = 2 * x & f . x > - 1 & f . x < 1 ) ; :: thesis: ( (1 / 2) (#) (arccos * f) is_differentiable_on Z & ( for x being Real st x in Z holds
(((1 / 2) (#) (arccos * f)) `| Z) . x = - (1 / (sqrt (1 - ((2 * x) ^2)))) ) )
A3: ( Z c= dom (arccos * f) & ( for x being Real st x in Z holds
( f . x = (2 * x) + 0 & f . x > - 1 & f . x < 1 ) ) ) by A1, A2, VALUED_1:def 5;
then A4: arccos * f is_differentiable_on Z by Th15;
for x being Real st x in Z holds
(((1 / 2) (#) (arccos * f)) `| Z) . x = - (1 / (sqrt (1 - ((2 * x) ^2))))

proof

let x be Real; :: thesis: ( x in Z implies (((1 / 2) (#) (arccos * f)) `| Z) . x = - (1 / (sqrt (1 - ((2 * x) ^2)))) )
assume A5: x in Z ; :: thesis: (((1 / 2) (#) (arccos * f)) `| Z) . x = - (1 / (sqrt (1 - ((2 * x) ^2))))
then (((1 / 2) (#) (arccos * f)) `| Z) . x = (1 / 2) * (diff ((arccos * f),x)) by A1, A4, FDIFF_1:20
.= (1 / 2) * (((arccos * f) `| Z) . x) by A4, A5, FDIFF_1:def 7
.= (1 / 2) * (- (2 / (sqrt (1 - (((2 * x) + 0) ^2))))) by A3, A5, Th15
.= - ((1 / 2) * (2 / (sqrt (1 - ((2 * x) ^2)))))
.= - (((1 / 2) * 2) / (sqrt (1 - ((2 * x) ^2)))) by XCMPLX_1:74
.= - (1 / (sqrt (1 - ((2 * x) ^2)))) ;
hence (((1 / 2) (#) (arccos * f)) `| Z) . x = - (1 / (sqrt (1 - ((2 * x) ^2)))) ; :: thesis: verum

end;

hence ( (1 / 2) (#) (arccos * f) is_differentiable_on Z & ( for x being Real st x in Z holds
(((1 / 2) (#) (arccos * f)) `| Z) . x = - (1 / (sqrt (1 - ((2 * x) ^2)))) ) ) by A1, A4, FDIFF_1:20; :: thesis: verum