let Z be open Subset of REAL; :: thesis: ( Z c= dom ((id Z) (#) arcsin) & Z c= ].(- 1),1.[ implies ( (id Z) (#) arcsin is_differentiable_on Z & ( for x being Real st x in Z holds
(((id Z) (#) arcsin) `| Z) . x = (arcsin . x) + (x / (sqrt (1 - (x ^2)))) ) ) )
assume that
A1: Z c= dom ((id Z) (#) arcsin) and
A2: Z c= ].(- 1),1.[ ; :: thesis: ( (id Z) (#) arcsin is_differentiable_on Z & ( for x being Real st x in Z holds
(((id Z) (#) arcsin) `| Z) . x = (arcsin . x) + (x / (sqrt (1 - (x ^2)))) ) )
A3: for x being Real st x in Z holds
(id Z) . x = (1 * x) + 0 by FUNCT_1:18;
Z c= (dom (id Z)) /\ (dom arcsin) by A1, VALUED_1:def 4;
then A4: Z c= dom (id Z) by XBOOLE_1:18;
then A5: id Z is_differentiable_on Z by A3, FDIFF_1:23;
A6: arcsin is_differentiable_on Z by A2, FDIFF_1:26, SIN_COS6:83;
for x being Real st x in Z holds
(((id Z) (#) arcsin) `| Z) . x = (arcsin . x) + (x / (sqrt (1 - (x ^2))))

proof

let x be Real; :: thesis: ( x in Z implies (((id Z) (#) arcsin) `| Z) . x = (arcsin . x) + (x / (sqrt (1 - (x ^2)))) )
assume A7: x in Z ; :: thesis: (((id Z) (#) arcsin) `| Z) . x = (arcsin . x) + (x / (sqrt (1 - (x ^2))))
then A8: ( - 1 < x & x < 1 ) by A2, XXREAL_1:4;
(((id Z) (#) arcsin) `| Z) . x = ((arcsin . x) * (diff ((id Z),x))) + (((id Z) . x) * (diff (arcsin,x))) by A1, A5, A6, A7, FDIFF_1:21
.= ((arcsin . x) * (((id Z) `| Z) . x)) + (((id Z) . x) * (diff (arcsin,x))) by A5, A7, FDIFF_1:def 7
.= ((arcsin . x) * 1) + (((id Z) . x) * (diff (arcsin,x))) by A4, A3, A7, FDIFF_1:23
.= ((arcsin . x) * 1) + (((id Z) . x) * (1 / (sqrt (1 - (x ^2))))) by A8, SIN_COS6:83
.= (arcsin . x) + (x * (1 / (sqrt (1 - (x ^2))))) by A7, FUNCT_1:18
.= (arcsin . x) + (x / (sqrt (1 - (x ^2)))) by XCMPLX_1:99 ;
hence (((id Z) (#) arcsin) `| Z) . x = (arcsin . x) + (x / (sqrt (1 - (x ^2)))) ; :: thesis: verum

end;

hence ( (id Z) (#) arcsin is_differentiable_on Z & ( for x being Real st x in Z holds
(((id Z) (#) arcsin) `| Z) . x = (arcsin . x) + (x / (sqrt (1 - (x ^2)))) ) ) by A1, A5, A6, FDIFF_1:21; :: thesis: verum