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

proof

let x be Real; :: thesis: ( x in Z implies ln * arccos is_differentiable_in x )
assume x in Z ; :: thesis: ln * arccos is_differentiable_in x
then ( arccos is_differentiable_in x & arccos . x > 0 ) by A2, A3, FDIFF_1:16, SIN_COS6:108;
hence ln * arccos is_differentiable_in x by TAYLOR_1:20; :: thesis: verum

end;

then A5: ln * arccos is_differentiable_on Z by A1, FDIFF_1:16;
for x being Real st x in Z holds
((ln * arccos ) `| Z) . x = - (1 / ((sqrt (1 - (x ^2 ))) * (arccos . x)))

proof

let x be Real; :: thesis: ( x in Z implies ((ln * arccos ) `| Z) . x = - (1 / ((sqrt (1 - (x ^2 ))) * (arccos . x))) )
assume A6: x in Z ; :: thesis: ((ln * arccos ) `| Z) . x = - (1 / ((sqrt (1 - (x ^2 ))) * (arccos . x)))
then A7: ( - 1 < x & x < 1 ) by A2, XXREAL_1:4;
( arccos is_differentiable_in x & arccos . x > 0 ) by A2, A3, A6, FDIFF_1:16, SIN_COS6:108;
then diff (ln * arccos ),x = (diff arccos ,x) / (arccos . x) by TAYLOR_1:20
.= (- (1 / (sqrt (1 - (x ^2 ))))) / (arccos . x) by A7, SIN_COS6:108
.= - ((1 / (sqrt (1 - (x ^2 )))) / (arccos . x)) by XCMPLX_1:188
.= - (1 / ((sqrt (1 - (x ^2 ))) * (arccos . x))) by XCMPLX_1:79 ;
hence ((ln * arccos ) `| Z) . x = - (1 / ((sqrt (1 - (x ^2 ))) * (arccos . x))) by A5, A6, FDIFF_1:def 8; :: thesis: verum

end;

hence ( ln * arccos is_differentiable_on Z & ( for x being Real st x in Z holds
((ln * arccos ) `| Z) . x = - (1 / ((sqrt (1 - (x ^2 ))) * (arccos . x))) ) ) by A1, A4, FDIFF_1:16; :: thesis: verum