let Z be open Subset of REAL; :: thesis: ( Z c= dom (ln * arcsin) & Z c= ].(- 1),1.[ & ( for x being Real st x in Z holds
arcsin . x > 0 ) implies ( ln * arcsin is_differentiable_on Z & ( for x being Real st x in Z holds
((ln * arcsin) `| Z) . x = 1 / ((sqrt (1 - (x ^2))) * (arcsin . x)) ) ) )
assume that
A1: Z c= dom (ln * arcsin) and
A2: Z c= ].(- 1),1.[ and
A3: for x being Real st x in Z holds
arcsin . x > 0 ; :: thesis: ( ln * arcsin is_differentiable_on Z & ( for x being Real st x in Z holds
((ln * arcsin) `| Z) . x = 1 / ((sqrt (1 - (x ^2))) * (arcsin . x)) ) )
A4: for x being Real st x in Z holds
ln * arcsin is_differentiable_in x

proof

let x be Real; :: thesis: ( x in Z implies ln * arcsin is_differentiable_in x )
assume x in Z ; :: thesis: ln * arcsin is_differentiable_in x
then ( arcsin is_differentiable_in x & arcsin . x > 0 ) by A2, A3, FDIFF_1:9, SIN_COS6:83;
hence ln * arcsin is_differentiable_in x by TAYLOR_1:20; :: thesis: verum

end;

then A5: ln * arcsin is_differentiable_on Z by A1, FDIFF_1:9;
for x being Real st x in Z holds
((ln * arcsin) `| Z) . x = 1 / ((sqrt (1 - (x ^2))) * (arcsin . x))

proof

let x be Real; :: thesis: ( x in Z implies ((ln * arcsin) `| Z) . x = 1 / ((sqrt (1 - (x ^2))) * (arcsin . x)) )
assume A6: x in Z ; :: thesis: ((ln * arcsin) `| Z) . x = 1 / ((sqrt (1 - (x ^2))) * (arcsin . x))
then A7: ( - 1 < x & x < 1 ) by A2, XXREAL_1:4;
( arcsin is_differentiable_in x & arcsin . x > 0 ) by A2, A3, A6, FDIFF_1:9, SIN_COS6:83;
then diff ((ln * arcsin),x) = (diff (arcsin,x)) / (arcsin . x) by TAYLOR_1:20
.= (1 / (sqrt (1 - (x ^2)))) / (arcsin . x) by A7, SIN_COS6:83
.= 1 / ((sqrt (1 - (x ^2))) * (arcsin . x)) by XCMPLX_1:78 ;
hence ((ln * arcsin) `| Z) . x = 1 / ((sqrt (1 - (x ^2))) * (arcsin . x)) by A5, A6, FDIFF_1:def 7; :: thesis: verum

end;

hence ( ln * arcsin is_differentiable_on Z & ( for x being Real st x in Z holds
((ln * arcsin) `| Z) . x = 1 / ((sqrt (1 - (x ^2))) * (arcsin . x)) ) ) by A1, A4, FDIFF_1:9; :: thesis: verum