let Z be open Subset of REAL; :: thesis: ( Z c= dom ((1 / 2) (#) ((#Z 2) * arccos)) & Z c= ].(- 1),1.[ implies ( (1 / 2) (#) ((#Z 2) * arccos) is_differentiable_on Z & ( for x being Real st x in Z holds
(((1 / 2) (#) ((#Z 2) * arccos)) `| Z) . x = - ((arccos . x) / (sqrt (1 - (x ^2)))) ) ) )
assume that
A1: Z c= dom ((1 / 2) (#) ((#Z 2) * arccos)) and
A2: Z c= ].(- 1),1.[ ; :: thesis: ( (1 / 2) (#) ((#Z 2) * arccos) is_differentiable_on Z & ( for x being Real st x in Z holds
(((1 / 2) (#) ((#Z 2) * arccos)) `| Z) . x = - ((arccos . x) / (sqrt (1 - (x ^2)))) ) )
A3: Z c= dom ((#Z 2) * arccos) by A1, VALUED_1:def 5;
then A4: (#Z 2) * arccos is_differentiable_on Z by A2, Th11;
for x being Real st x in Z holds
(((1 / 2) (#) ((#Z 2) * arccos)) `| Z) . x = - ((arccos . x) / (sqrt (1 - (x ^2))))

proof

let x be Real; :: thesis: ( x in Z implies (((1 / 2) (#) ((#Z 2) * arccos)) `| Z) . x = - ((arccos . x) / (sqrt (1 - (x ^2)))) )
assume A5: x in Z ; :: thesis: (((1 / 2) (#) ((#Z 2) * arccos)) `| Z) . x = - ((arccos . x) / (sqrt (1 - (x ^2))))
then (((1 / 2) (#) ((#Z 2) * arccos)) `| Z) . x = (1 / 2) * (diff (((#Z 2) * arccos),x)) by A1, A4, FDIFF_1:20
.= (1 / 2) * ((((#Z 2) * arccos) `| Z) . x) by A4, A5, FDIFF_1:def 7
.= (1 / 2) * (- ((2 * ((arccos . x) #Z (2 - 1))) / (sqrt (1 - (x ^2))))) by A2, A3, A5, Th11
.= - ((1 / 2) * ((2 * ((arccos . x) #Z (2 - 1))) / (sqrt (1 - (x ^2)))))
.= - (((1 / 2) * (2 * ((arccos . x) #Z (2 - 1)))) / (sqrt (1 - (x ^2)))) by XCMPLX_1:74
.= - ((arccos . x) / (sqrt (1 - (x ^2)))) by PREPOWER:35 ;
hence (((1 / 2) (#) ((#Z 2) * arccos)) `| Z) . x = - ((arccos . x) / (sqrt (1 - (x ^2)))) ; :: thesis: verum

end;

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