let Z be open Subset of REAL ; :: thesis: ( Z c= dom ((1 / 2) (#) ((#Z 2) * arcsin )) & Z c= ].(- 1),1.[ implies ( (1 / 2) (#) ((#Z 2) * arcsin ) is_differentiable_on Z & ( for x being Real st x in Z holds
(((1 / 2) (#) ((#Z 2) * arcsin )) `| Z) . x = (arcsin . x) / (sqrt (1 - (x ^2 ))) ) ) )
assume that
A1: Z c= dom ((1 / 2) (#) ((#Z 2) * arcsin )) and
A2: Z c= ].(- 1),1.[ ; :: thesis: ( (1 / 2) (#) ((#Z 2) * arcsin ) is_differentiable_on Z & ( for x being Real st x in Z holds
(((1 / 2) (#) ((#Z 2) * arcsin )) `| Z) . x = (arcsin . x) / (sqrt (1 - (x ^2 ))) ) )
A3: Z c= dom ((#Z 2) * arcsin ) by A1, VALUED_1:def 5;
then A4: (#Z 2) * arcsin is_differentiable_on Z by A2, Th10;
for x being Real st x in Z holds
(((1 / 2) (#) ((#Z 2) * arcsin )) `| Z) . x = (arcsin . x) / (sqrt (1 - (x ^2 )))

proof

let x be Real; :: thesis: ( x in Z implies (((1 / 2) (#) ((#Z 2) * arcsin )) `| Z) . x = (arcsin . x) / (sqrt (1 - (x ^2 ))) )
assume A5: x in Z ; :: thesis: (((1 / 2) (#) ((#Z 2) * arcsin )) `| Z) . x = (arcsin . x) / (sqrt (1 - (x ^2 )))
then (((1 / 2) (#) ((#Z 2) * arcsin )) `| Z) . x = (1 / 2) * (diff ((#Z 2) * arcsin ),x) by A1, A4, FDIFF_1:28
.= (1 / 2) * ((((#Z 2) * arcsin ) `| Z) . x) by A4, A5, FDIFF_1:def 8
.= (1 / 2) * ((2 * ((arcsin . x) #Z (2 - 1))) / (sqrt (1 - (x ^2 )))) by A2, A3, A5, Th10
.= ((1 / 2) * (2 * ((arcsin . x) #Z (2 - 1)))) / (sqrt (1 - (x ^2 ))) by XCMPLX_1:75
.= (arcsin . x) / (sqrt (1 - (x ^2 ))) by PREPOWER:45 ;
hence (((1 / 2) (#) ((#Z 2) * arcsin )) `| Z) . x = (arcsin . x) / (sqrt (1 - (x ^2 ))) ; :: thesis: verum

end;

hence ( (1 / 2) (#) ((#Z 2) * arcsin ) is_differentiable_on Z & ( for x being Real st x in Z holds
(((1 / 2) (#) ((#Z 2) * arcsin )) `| Z) . x = (arcsin . x) / (sqrt (1 - (x ^2 ))) ) ) by A1, A4, FDIFF_1:28; :: thesis: verum