let r be Real; :: thesis: for Z being open Subset of REAL st Z c= ].(- 1),1.[ & Z c= dom (r (#) arcsin ) holds
( r (#) arcsin is_differentiable_on Z & ( for x being Real st x in Z holds
((r (#) arcsin ) `| Z) . x = r / (sqrt (1 - (x ^2 ))) ) )
let Z be open Subset of REAL ; :: thesis: ( Z c= ].(- 1),1.[ & Z c= dom (r (#) arcsin ) implies ( r (#) arcsin is_differentiable_on Z & ( for x being Real st x in Z holds
((r (#) arcsin ) `| Z) . x = r / (sqrt (1 - (x ^2 ))) ) ) )
assume that
A1: Z c= ].(- 1),1.[ and
A2: Z c= dom (r (#) arcsin ) ; :: thesis: ( r (#) arcsin is_differentiable_on Z & ( for x being Real st x in Z holds
((r (#) arcsin ) `| Z) . x = r / (sqrt (1 - (x ^2 ))) ) )
A3: arcsin is_differentiable_on Z by A1, FDIFF_1:34, SIN_COS6:84;
for x being Real st x in Z holds
((r (#) arcsin ) `| Z) . x = r / (sqrt (1 - (x ^2 )))

proof

let x be Real; :: thesis: ( x in Z implies ((r (#) arcsin ) `| Z) . x = r / (sqrt (1 - (x ^2 ))) )
assume A4: x in Z ; :: thesis: ((r (#) arcsin ) `| Z) . x = r / (sqrt (1 - (x ^2 )))
then A5: ( - 1 < x & x < 1 ) by A1, XXREAL_1:4;
((r (#) arcsin ) `| Z) . x = r * (diff arcsin ,x) by A2, A3, A4, FDIFF_1:28
.= r * (1 / (sqrt (1 - (x ^2 )))) by A5, SIN_COS6:84
.= r / (sqrt (1 - (x ^2 ))) by XCMPLX_1:100 ;
hence ((r (#) arcsin ) `| Z) . x = r / (sqrt (1 - (x ^2 ))) ; :: thesis: verum

end;

hence ( r (#) arcsin is_differentiable_on Z & ( for x being Real st x in Z holds
((r (#) arcsin ) `| Z) . x = r / (sqrt (1 - (x ^2 ))) ) ) by A2, A3, FDIFF_1:28; :: thesis: verum