let Z be open Subset of REAL; :: thesis: ( Z c= dom (2 (#) ((#R (1 / 2)) * arctan)) & Z c= ].(- 1),1.[ & ( for x being Real st x in Z holds
arctan . x > 0 ) implies ( 2 (#) ((#R (1 / 2)) * arctan) is_differentiable_on Z & ( for x being Real st x in Z holds
((2 (#) ((#R (1 / 2)) * arctan)) `| Z) . x = ((arctan . x) #R (- (1 / 2))) / (1 + (x ^2)) ) ) )
assume that
A1: Z c= dom (2 (#) ((#R (1 / 2)) * arctan)) and
A2: Z c= ].(- 1),1.[ and
A3: for x being Real st x in Z holds
arctan . x > 0 ; :: thesis: ( 2 (#) ((#R (1 / 2)) * arctan) is_differentiable_on Z & ( for x being Real st x in Z holds
((2 (#) ((#R (1 / 2)) * arctan)) `| Z) . x = ((arctan . x) #R (- (1 / 2))) / (1 + (x ^2)) ) )
A4: for x being Real st x in Z holds
(#R (1 / 2)) * arctan is_differentiable_in x

proof

let x be Real; :: thesis: ( x in Z implies (#R (1 / 2)) * arctan is_differentiable_in x )
assume A5: x in Z ; :: thesis: (#R (1 / 2)) * arctan is_differentiable_in x
then A6: arctan . x > 0 by A3;
arctan is_differentiable_on Z by A2, SIN_COS9:81;
then arctan is_differentiable_in x by A5, FDIFF_1:9;
hence (#R (1 / 2)) * arctan is_differentiable_in x by A6, TAYLOR_1:22; :: thesis: verum

end;

Z c= dom ((#R (1 / 2)) * arctan) by A1, VALUED_1:def 5;
then A7: (#R (1 / 2)) * arctan is_differentiable_on Z by A4, FDIFF_1:9;
for x being Real st x in Z holds
((2 (#) ((#R (1 / 2)) * arctan)) `| Z) . x = ((arctan . x) #R (- (1 / 2))) / (1 + (x ^2))

proof

let x be Real; :: thesis: ( x in Z implies ((2 (#) ((#R (1 / 2)) * arctan)) `| Z) . x = ((arctan . x) #R (- (1 / 2))) / (1 + (x ^2)) )
assume A8: x in Z ; :: thesis: ((2 (#) ((#R (1 / 2)) * arctan)) `| Z) . x = ((arctan . x) #R (- (1 / 2))) / (1 + (x ^2))
then A9: arctan . x > 0 by A3;
A10: arctan is_differentiable_on Z by A2, SIN_COS9:81;
then A11: arctan is_differentiable_in x by A8, FDIFF_1:9;
((2 (#) ((#R (1 / 2)) * arctan)) `| Z) . x = 2 * (diff (((#R (1 / 2)) * arctan),x)) by A1, A7, A8, FDIFF_1:20
.= 2 * (((1 / 2) * ((arctan . x) #R ((1 / 2) - 1))) * (diff (arctan,x))) by A11, A9, TAYLOR_1:22
.= 2 * (((1 / 2) * ((arctan . x) #R ((1 / 2) - 1))) * ((arctan `| Z) . x)) by A8, A10, FDIFF_1:def 7
.= 2 * (((1 / 2) * ((arctan . x) #R ((1 / 2) - 1))) * (1 / (1 + (x ^2)))) by A2, A8, SIN_COS9:81
.= ((arctan . x) #R (- (1 / 2))) / (1 + (x ^2)) ;
hence ((2 (#) ((#R (1 / 2)) * arctan)) `| Z) . x = ((arctan . x) #R (- (1 / 2))) / (1 + (x ^2)) ; :: thesis: verum

end;

hence ( 2 (#) ((#R (1 / 2)) * arctan) is_differentiable_on Z & ( for x being Real st x in Z holds
((2 (#) ((#R (1 / 2)) * arctan)) `| Z) . x = ((arctan . x) #R (- (1 / 2))) / (1 + (x ^2)) ) ) by A1, A7, FDIFF_1:20; :: thesis: verum