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

proof

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

end;

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