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 & f2 = #Z 2 & ( 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 ) )
A4: ( f = f1 - f2 & f2 = #Z 2 & ( for x being Real st x in Z holds
( f1 . x = 1 & f . x > 0 ) ) ) by A3;
A5: Z c= (dom ((id Z) (#) arccos )) /\ (dom ((#R (1 / 2)) * f)) by A1, VALUED_1:12;
then A6: Z c= dom ((id Z) (#) arccos ) by XBOOLE_1:18;
A7: Z c= dom ((#R (1 / 2)) * f) by A5, XBOOLE_1:18;
A8: ( (id Z) (#) arccos is_differentiable_on Z & ( for x being Real st x in Z holds
(((id Z) (#) arccos ) `| Z) . x = (arccos . x) - (x / (sqrt (1 - (x ^2 )))) ) ) by A2, A6, Th17;
A9: ( (#R (1 / 2)) * f is_differentiable_on Z & ( for x being Real st x in Z holds
(((#R (1 / 2)) * f) `| Z) . x = - (x * ((1 - (x #Z 2)) #R (- (1 / 2)))) ) ) by A4, A7, Th22;
A10: 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 A11: 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 (f1 - f2) . x = (f1 . x) - (f2 . x) by VALUED_1:13
.= 1 - (f2 . x) by A3, A11
.= 1 - (x #Z 2) by A3, TAYLOR_1:def 1 ;
then ( f . x = 1 - (x #Z 2) & f . x > 0 ) by A3, A11;
then A12: 1 - (x ^2 ) > 0 by Th1;
(1 - (x #Z 2)) #R (- (1 / 2)) = (1 - (x ^2 )) #R (- (1 / 2)) by Th1
.= 1 / (sqrt (1 - (x ^2 ))) by A12, 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 A13: 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, A8, A9, FDIFF_1:27
.= ((((id Z) (#) arccos ) `| Z) . x) - (diff ((#R (1 / 2)) * f),x) by A8, A13, FDIFF_1:def 8
.= ((((id Z) (#) arccos ) `| Z) . x) - ((((#R (1 / 2)) * f) `| Z) . x) by A9, A13, FDIFF_1:def 8
.= ((arccos . x) - (x / (sqrt (1 - (x ^2 ))))) - ((((#R (1 / 2)) * f) `| Z) . x) by A2, A6, A13, Th17
.= ((arccos . x) - (x / (sqrt (1 - (x ^2 ))))) - (- (x * ((1 - (x #Z 2)) #R (- (1 / 2))))) by A4, A7, A13, 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 A10, A13
.= 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, A8, A9, FDIFF_1:27; :: thesis: verum