let r be Real; :: thesis: for Z being open Subset of REAL st Z c= ].(- 1),1.[ holds
( r (#) arctan is_differentiable_on Z & ( for x being Real st x in Z holds
((r (#) arctan) `| Z) . x = r / (1 + (x ^2)) ) )
let Z be open Subset of REAL; :: thesis: ( Z c= ].(- 1),1.[ implies ( r (#) arctan is_differentiable_on Z & ( for x being Real st x in Z holds
((r (#) arctan) `| Z) . x = r / (1 + (x ^2)) ) ) )
assume A1: Z c= ].(- 1),1.[ ; :: thesis: ( r (#) arctan is_differentiable_on Z & ( for x being Real st x in Z holds
((r (#) arctan) `| Z) . x = r / (1 + (x ^2)) ) )
].(- 1),1.[ c= [.(- 1),1.] by XXREAL_1:25;
then ].(- 1),1.[ c= dom arctan by Th23;
then Z c= dom arctan by A1;
then A2: Z c= dom (r (#) arctan) by VALUED_1:def 5;
A3: arctan is_differentiable_on Z by A1, Th81;
for x being Real st x in Z holds
((r (#) arctan) `| Z) . x = r / (1 + (x ^2))

proof

let x be Real; :: thesis: ( x in Z implies ((r (#) arctan) `| Z) . x = r / (1 + (x ^2)) )
assume A4: x in Z ; :: thesis: ((r (#) arctan) `| Z) . x = r / (1 + (x ^2))
then A5: - 1 < x by A1, XXREAL_1:4;
A6: x < 1 by A1, A4, XXREAL_1:4;
((r (#) arctan) `| Z) . x = r * (diff (arctan,x)) by A2, A3, A4, FDIFF_1:20
.= r * (1 / (1 + (x ^2))) by A5, A6, Th75
.= r / (1 + (x ^2)) ;
hence ((r (#) arctan) `| Z) . x = r / (1 + (x ^2)) ; :: thesis: verum

end;

hence ( r (#) arctan is_differentiable_on Z & ( for x being Real st x in Z holds
((r (#) arctan) `| Z) . x = r / (1 + (x ^2)) ) ) by A2, A3, FDIFF_1:20; :: thesis: verum