let Z be open Subset of REAL ; :: thesis: for f, f1, f2 being PartFunc of REAL ,REAL st Z c= dom (((id Z) (#) arccos ) - ((#R (1 / 2)) * f)) & Z c= ].(- 1),1.[ & f = f1 - f2 & f2 = #Z 2 & ( for x being Real st x in Z holds
( f1 . x = 1 & f . x > 0 & x <> 0 ) ) holds
( ((id Z) (#) arccos ) - ((#R (1 / 2)) * f) is_differentiable_on Z & ( for x being Real st x in Z holds
((((id Z) (#) arccos ) - ((#R (1 / 2)) * f)) `| Z) . x = arccos . x ) )
let f, f1, f2 be PartFunc of REAL ,REAL ; :: thesis: ( Z c= dom (((id Z) (#) arccos ) - ((#R (1 / 2)) * f)) & Z c= ].(- 1),1.[ & f = f1 - f2 & f2 = #Z 2 & ( for x being Real st x in Z holds
( f1 . x = 1 & f . x > 0 & x <> 0 ) ) implies ( ((id Z) (#) arccos ) - ((#R (1 / 2)) * f) is_differentiable_on Z & ( for x being Real st x in Z holds
((((id Z) (#) arccos ) - ((#R (1 / 2)) * f)) `| Z) . x = arccos . x ) ) )
assume that
A1: Z c= dom (((id Z) (#) arccos ) - ((#R (1 / 2)) * f)) and
A2: Z c= ].(- 1),1.[ and
A3: f = f1 - f2 and
A4: f2 = #Z 2 and
A5: for x being Real st x in Z holds
( f1 . x = 1 & f . x > 0 & x <> 0 ) ; :: thesis: ( ((id Z) (#) arccos ) - ((#R (1 / 2)) * f) is_differentiable_on Z & ( for x being Real st x in Z holds
((((id Z) (#) arccos ) - ((#R (1 / 2)) * f)) `| Z) . x = arccos . x ) )
A6: Z c= (dom ((id Z) (#) arccos )) /\ (dom ((#R (1 / 2)) * f)) by A1, VALUED_1:12;
then A7: Z c= dom ((#R (1 / 2)) * f) by XBOOLE_1:18;
A8: Z c= dom ((id Z) (#) arccos ) by A6, XBOOLE_1:18;
then A9: (id Z) (#) arccos is_differentiable_on Z by A2, Th17;
A10: for x being Real st x in Z holds
( f1 . x = 1 & f . x > 0 ) by A5;
then A11: (#R (1 / 2)) * f is_differentiable_on Z by A3, A4, A7, Th22;
A12: for x being Real st x in Z holds
x * ((1 - (x #Z 2)) #R (- (1 / 2))) = x / (sqrt (1 - (x ^2 )))

proof

let x be Real; :: thesis: ( x in Z implies x * ((1 - (x #Z 2)) #R (- (1 / 2))) = x / (sqrt (1 - (x ^2 ))) )
assume A13: x in Z ; :: thesis: x * ((1 - (x #Z 2)) #R (- (1 / 2))) = x / (sqrt (1 - (x ^2 )))
then x in dom (f1 - f2) by A3, A7, FUNCT_1:21;
then A14: (f1 - f2) . x = (f1 . x) - (f2 . x) by VALUED_1:13
.= 1 - (f2 . x) by A5, A13
.= 1 - (x #Z 2) by A4, TAYLOR_1:def 1 ;
f . x > 0 by A5, A13;
then A15: 1 - (x ^2 ) > 0 by A3, A14, Th1;
(1 - (x #Z 2)) #R (- (1 / 2)) = (1 - (x ^2 )) #R (- (1 / 2)) by Th1
.= 1 / (sqrt (1 - (x ^2 ))) by A15, Th3 ;
hence x * ((1 - (x #Z 2)) #R (- (1 / 2))) = x / (sqrt (1 - (x ^2 ))) by XCMPLX_1:100; :: thesis: verum

end;

for x being Real st x in Z holds
((((id Z) (#) arccos ) - ((#R (1 / 2)) * f)) `| Z) . x = arccos . x

proof

let x be Real; :: thesis: ( x in Z implies ((((id Z) (#) arccos ) - ((#R (1 / 2)) * f)) `| Z) . x = arccos . x )
assume A16: x in Z ; :: thesis: ((((id Z) (#) arccos ) - ((#R (1 / 2)) * f)) `| Z) . x = arccos . x
hence ((((id Z) (#) arccos ) - ((#R (1 / 2)) * f)) `| Z) . x = (diff ((id Z) (#) arccos ),x) - (diff ((#R (1 / 2)) * f),x) by A1, A9, A11, FDIFF_1:27
.= ((((id Z) (#) arccos ) `| Z) . x) - (diff ((#R (1 / 2)) * f),x) by A9, A16, FDIFF_1:def 8
.= ((((id Z) (#) arccos ) `| Z) . x) - ((((#R (1 / 2)) * f) `| Z) . x) by A11, A16, FDIFF_1:def 8
.= ((arccos . x) - (x / (sqrt (1 - (x ^2 ))))) - ((((#R (1 / 2)) * f) `| Z) . x) by A2, A8, A16, Th17
.= ((arccos . x) - (x / (sqrt (1 - (x ^2 ))))) - (- (x * ((1 - (x #Z 2)) #R (- (1 / 2))))) by A3, A4, A10, A7, A16, Th22
.= ((arccos . x) - (x / (sqrt (1 - (x ^2 ))))) + (x * ((1 - (x #Z 2)) #R (- (1 / 2))))
.= ((arccos . x) - (x / (sqrt (1 - (x ^2 ))))) + (x / (sqrt (1 - (x ^2 )))) by A12, A16
.= arccos . x ;
:: thesis: verum

end;

hence ( ((id Z) (#) arccos ) - ((#R (1 / 2)) * f) is_differentiable_on Z & ( for x being Real st x in Z holds
((((id Z) (#) arccos ) - ((#R (1 / 2)) * f)) `| Z) . x = arccos . x ) ) by A1, A9, A11, FDIFF_1:27; :: thesis: verum