let a, b be Real; :: thesis: for Z being open Subset of REAL
for f being PartFunc of REAL ,REAL st Z c= dom (f (#) arccos ) & Z c= ].(- 1),1.[ & ( for x being Real st x in Z holds
f . x = (a * x) + b ) holds
( f (#) arccos is_differentiable_on Z & ( for x being Real st x in Z holds
((f (#) arccos ) `| Z) . x = (a * (arccos . x)) - (((a * x) + b) / (sqrt (1 - (x ^2 )))) ) )
let Z be open Subset of REAL ; :: thesis: for f being PartFunc of REAL ,REAL st Z c= dom (f (#) arccos ) & Z c= ].(- 1),1.[ & ( for x being Real st x in Z holds
f . x = (a * x) + b ) holds
( f (#) arccos is_differentiable_on Z & ( for x being Real st x in Z holds
((f (#) arccos ) `| Z) . x = (a * (arccos . x)) - (((a * x) + b) / (sqrt (1 - (x ^2 )))) ) )
let f be PartFunc of REAL ,REAL ; :: thesis: ( Z c= dom (f (#) arccos ) & Z c= ].(- 1),1.[ & ( for x being Real st x in Z holds
f . x = (a * x) + b ) implies ( f (#) arccos is_differentiable_on Z & ( for x being Real st x in Z holds
((f (#) arccos ) `| Z) . x = (a * (arccos . x)) - (((a * x) + b) / (sqrt (1 - (x ^2 )))) ) ) )
assume that
A1: Z c= dom (f (#) arccos ) and
A2: Z c= ].(- 1),1.[ and
A3: for x being Real st x in Z holds
f . x = (a * x) + b ; :: thesis: ( f (#) arccos is_differentiable_on Z & ( for x being Real st x in Z holds
((f (#) arccos ) `| Z) . x = (a * (arccos . x)) - (((a * x) + b) / (sqrt (1 - (x ^2 )))) ) )
Z c= (dom f) /\ (dom arccos ) by A1, VALUED_1:def 4;
then A4: Z c= dom f by XBOOLE_1:18;
then A5: ( f is_differentiable_on Z & ( for x being Real st x in Z holds
(f `| Z) . x = a ) ) by A3, FDIFF_1:31;
A6: arccos is_differentiable_on Z by A2, FDIFF_1:34, SIN_COS6:108;
for x being Real st x in Z holds
((f (#) arccos ) `| Z) . x = (a * (arccos . x)) - (((a * x) + b) / (sqrt (1 - (x ^2 ))))

proof

let x be Real; :: thesis: ( x in Z implies ((f (#) arccos ) `| Z) . x = (a * (arccos . x)) - (((a * x) + b) / (sqrt (1 - (x ^2 )))) )
assume A7: x in Z ; :: thesis: ((f (#) arccos ) `| Z) . x = (a * (arccos . x)) - (((a * x) + b) / (sqrt (1 - (x ^2 ))))
then A8: ( - 1 < x & x < 1 ) by A2, XXREAL_1:4;
((f (#) arccos ) `| Z) . x = ((arccos . x) * (diff f,x)) + ((f . x) * (diff arccos ,x)) by A1, A5, A6, A7, FDIFF_1:29
.= ((arccos . x) * ((f `| Z) . x)) + ((f . x) * (diff arccos ,x)) by A5, A7, FDIFF_1:def 8
.= ((arccos . x) * a) + ((f . x) * (diff arccos ,x)) by A3, A4, A7, FDIFF_1:31
.= ((arccos . x) * a) + ((f . x) * (- (1 / (sqrt (1 - (x ^2 )))))) by A8, SIN_COS6:108
.= ((arccos . x) * a) - ((f . x) * (1 / (sqrt (1 - (x ^2 )))))
.= (a * (arccos . x)) - (((a * x) + b) * (1 / (sqrt (1 - (x ^2 ))))) by A3, A7
.= (a * (arccos . x)) - (((a * x) + b) / (sqrt (1 - (x ^2 )))) by XCMPLX_1:100 ;
hence ((f (#) arccos ) `| Z) . x = (a * (arccos . x)) - (((a * x) + b) / (sqrt (1 - (x ^2 )))) ; :: thesis: verum

end;

hence ( f (#) arccos is_differentiable_on Z & ( for x being Real st x in Z holds
((f (#) arccos ) `| Z) . x = (a * (arccos . x)) - (((a * x) + b) / (sqrt (1 - (x ^2 )))) ) ) by A1, A5, A6, FDIFF_1:29; :: thesis: verum