let r, s be Real; :: thesis: for Z being open Subset of REAL
for f being PartFunc of REAL ,REAL st Z c= dom (f (#) arctan ) & Z c= ].(- 1),1.[ & ( for x being Real st x in Z holds
f . x = (r * x) + s ) holds
( f (#) arctan is_differentiable_on Z & ( for x being Real st x in Z holds
((f (#) arctan ) `| Z) . x = (r * (arctan . x)) + (((r * x) + s) / (1 + (x ^2 ))) ) )
let Z be open Subset of REAL ; :: thesis: for f being PartFunc of REAL ,REAL st Z c= dom (f (#) arctan ) & Z c= ].(- 1),1.[ & ( for x being Real st x in Z holds
f . x = (r * x) + s ) holds
( f (#) arctan is_differentiable_on Z & ( for x being Real st x in Z holds
((f (#) arctan ) `| Z) . x = (r * (arctan . x)) + (((r * x) + s) / (1 + (x ^2 ))) ) )
let f be PartFunc of REAL ,REAL ; :: thesis: ( Z c= dom (f (#) arctan ) & Z c= ].(- 1),1.[ & ( for x being Real st x in Z holds
f . x = (r * x) + s ) implies ( f (#) arctan is_differentiable_on Z & ( for x being Real st x in Z holds
((f (#) arctan ) `| Z) . x = (r * (arctan . x)) + (((r * x) + s) / (1 + (x ^2 ))) ) ) )
assume that
A1: Z c= dom (f (#) arctan ) and
A2: Z c= ].(- 1),1.[ and
A3: for x being Real st x in Z holds
f . x = (r * x) + s ; :: thesis: ( f (#) arctan is_differentiable_on Z & ( for x being Real st x in Z holds
((f (#) arctan ) `| Z) . x = (r * (arctan . x)) + (((r * x) + s) / (1 + (x ^2 ))) ) )
Z c= (dom f) /\ (dom arctan ) by A1, VALUED_1:def 4;
then A4: Z c= dom f by XBOOLE_1:18;
then A5: f is_differentiable_on Z by A3, FDIFF_1:31;
A6: arctan is_differentiable_on Z by A2, Th81;
for x being Real st x in Z holds
((f (#) arctan ) `| Z) . x = (r * (arctan . x)) + (((r * x) + s) / (1 + (x ^2 )))

proof

let x be Real; :: thesis: ( x in Z implies ((f (#) arctan ) `| Z) . x = (r * (arctan . x)) + (((r * x) + s) / (1 + (x ^2 ))) )
assume A7: x in Z ; :: thesis: ((f (#) arctan ) `| Z) . x = (r * (arctan . x)) + (((r * x) + s) / (1 + (x ^2 )))
then A8: - 1 < x by A2, XXREAL_1:4;
A9: x < 1 by A2, A7, XXREAL_1:4;
((f (#) arctan ) `| Z) . x = ((arctan . x) * (diff f,x)) + ((f . x) * (diff arctan ,x)) by A1, A5, A6, A7, FDIFF_1:29
.= ((arctan . x) * ((f `| Z) . x)) + ((f . x) * (diff arctan ,x)) by A5, A7, FDIFF_1:def 8
.= ((arctan . x) * r) + ((f . x) * (diff arctan ,x)) by A3, A4, A7, FDIFF_1:31
.= ((arctan . x) * r) + ((f . x) * (1 / (1 + (x ^2 )))) by A8, A9, Th75
.= (r * (arctan . x)) + (((r * x) + s) / (1 + (x ^2 ))) by A3, A7 ;
hence ((f (#) arctan ) `| Z) . x = (r * (arctan . x)) + (((r * x) + s) / (1 + (x ^2 ))) ; :: thesis: verum

end;

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