let Z be open Subset of REAL ; :: thesis: ( Z c= dom ((1 / 2) (#) ((#Z 2) * arccos )) & Z c= ].(- 1),1.[ implies ( (1 / 2) (#) ((#Z 2) * arccos ) is_differentiable_on Z & ( for x being Real st x in Z holds
(((1 / 2) (#) ((#Z 2) * arccos )) `| Z) . x = - ((arccos . x) / (sqrt (1 - (x ^2 )))) ) ) )
assume that
A1: Z c= dom ((1 / 2) (#) ((#Z 2) * arccos )) and
A2: Z c= ].(- 1),1.[ ; :: thesis: ( (1 / 2) (#) ((#Z 2) * arccos ) is_differentiable_on Z & ( for x being Real st x in Z holds
(((1 / 2) (#) ((#Z 2) * arccos )) `| Z) . x = - ((arccos . x) / (sqrt (1 - (x ^2 )))) ) )
A3: Z c= dom ((#Z 2) * arccos ) by A1, VALUED_1:def 5;
then A4: ( (#Z 2) * arccos is_differentiable_on Z & ( for x being Real st x in Z holds
(((#Z 2) * arccos ) `| Z) . x = - ((2 * ((arccos . x) #Z (2 - 1))) / (sqrt (1 - (x ^2 )))) ) ) by A2, Th11;
for x being Real st x in Z holds
(((1 / 2) (#) ((#Z 2) * arccos )) `| Z) . x = - ((arccos . x) / (sqrt (1 - (x ^2 ))))

proof

let x be Real; :: thesis: ( x in Z implies (((1 / 2) (#) ((#Z 2) * arccos )) `| Z) . x = - ((arccos . x) / (sqrt (1 - (x ^2 )))) )
assume A5: x in Z ; :: thesis: (((1 / 2) (#) ((#Z 2) * arccos )) `| Z) . x = - ((arccos . x) / (sqrt (1 - (x ^2 ))))
then (((1 / 2) (#) ((#Z 2) * arccos )) `| Z) . x = (1 / 2) * (diff ((#Z 2) * arccos ),x) by A1, A4, FDIFF_1:28
.= (1 / 2) * ((((#Z 2) * arccos ) `| Z) . x) by A4, A5, FDIFF_1:def 8
.= (1 / 2) * (- ((2 * ((arccos . x) #Z (2 - 1))) / (sqrt (1 - (x ^2 ))))) by A2, A3, A5, Th11
.= - ((1 / 2) * ((2 * ((arccos . x) #Z (2 - 1))) / (sqrt (1 - (x ^2 )))))
.= - (((1 / 2) * (2 * ((arccos . x) #Z (2 - 1)))) / (sqrt (1 - (x ^2 )))) by XCMPLX_1:75
.= - ((arccos . x) / (sqrt (1 - (x ^2 )))) by PREPOWER:45 ;
hence (((1 / 2) (#) ((#Z 2) * arccos )) `| Z) . x = - ((arccos . x) / (sqrt (1 - (x ^2 )))) ; :: thesis: verum

end;

hence ( (1 / 2) (#) ((#Z 2) * arccos ) is_differentiable_on Z & ( for x being Real st x in Z holds
(((1 / 2) (#) ((#Z 2) * arccos )) `| Z) . x = - ((arccos . x) / (sqrt (1 - (x ^2 )))) ) ) by A1, A4, FDIFF_1:28; :: thesis: verum