let Z be open Subset of REAL; :: thesis: ( Z c= ].(- 1),1.[ implies ( arctan is_differentiable_on Z & ( for x being Real st x in Z holds
(arctan `| Z) . x = 1 / (1 + (x ^2)) ) ) )
assume A1: Z c= ].(- 1),1.[ ; :: thesis: ( arctan is_differentiable_on Z & ( for x being Real st x in Z holds
(arctan `| Z) . x = 1 / (1 + (x ^2)) ) )
then A2: arctan is_differentiable_on Z by Th73, FDIFF_1:26;
for x being Real st x in Z holds
(arctan `| Z) . x = 1 / (1 + (x ^2))

proof

let x be Real; :: thesis: ( x in Z implies (arctan `| Z) . x = 1 / (1 + (x ^2)) )
assume A3: x in Z ; :: thesis: (arctan `| Z) . x = 1 / (1 + (x ^2))
then A4: - 1 <= x by A1, XXREAL_1:4;
A5: x <= 1 by A1, A3, XXREAL_1:4;
thus (arctan `| Z) . x = diff (arctan,x) by A2, A3, FDIFF_1:def 7
.= 1 / (1 + (x ^2)) by A4, A5, Th75 ; :: thesis: verum

end;

hence ( arctan is_differentiable_on Z & ( for x being Real st x in Z holds
(arctan `| Z) . x = 1 / (1 + (x ^2)) ) ) by A1, Th73, FDIFF_1:26; :: thesis: verum