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) by A1, VALUED_1:def 5;
A4: for x being Real st x in Z holds
( f . x = (2 * x) + 0 & f . x > - 1 & f . x < 1 ) by A2;
then A5: ( arccos * f is_differentiable_on Z & ( for x being Real st x in Z holds
((arccos * f) `| Z) . x = - (2 / (sqrt (1 - (((2 * x) + 0 ) ^2 )))) ) ) by A3, 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 A6: 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, A5, FDIFF_1:28
.= (1 / 2) * (((arccos * f) `| Z) . x) by A5, A6, FDIFF_1:def 8
.= (1 / 2) * (- (2 / (sqrt (1 - (((2 * x) + 0 ) ^2 ))))) by A3, A4, A6, Th15
.= - ((1 / 2) * (2 / (sqrt (1 - ((2 * x) ^2 )))))
.= - (((1 / 2) * 2) / (sqrt (1 - ((2 * x) ^2 )))) by XCMPLX_1:75
.= - (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, A5, FDIFF_1:28; :: thesis: verum