let Z be open Subset of REAL; :: thesis: ( Z c= dom (arctan - (id Z)) & Z c= ].(- 1),1.[ implies ( arctan - (id Z) is_differentiable_on Z & ( for x being Real st x in Z holds
((arctan - (id Z)) `| Z) . x = - ((x ^2) / (1 + (x ^2))) ) ) )
assume that
A1: Z c= dom (arctan - (id Z)) and
A2: Z c= ].(- 1),1.[ ; :: thesis: ( arctan - (id Z) is_differentiable_on Z & ( for x being Real st x in Z holds
((arctan - (id Z)) `| Z) . x = - ((x ^2) / (1 + (x ^2))) ) )
A3: for x being Real st x in Z holds
(id Z) . x = (1 * x) + 0 by FUNCT_1:18;
Z c= (dom arctan) /\ (dom (id Z)) by A1, VALUED_1:12;
then A4: Z c= dom (id Z) by XBOOLE_1:18;
then A5: id Z is_differentiable_on Z by A3, FDIFF_1:23;
A6: arctan is_differentiable_on Z by A2, Th81;
for x being Real st x in Z holds
((arctan - (id Z)) `| Z) . x = - ((x ^2) / (1 + (x ^2)))

proof

let x be Real; :: thesis: ( x in Z implies ((arctan - (id Z)) `| Z) . x = - ((x ^2) / (1 + (x ^2))) )
A7: 1 + (x ^2) > 0 by XREAL_1:34, XREAL_1:63;
assume A8: x in Z ; :: thesis: ((arctan - (id Z)) `| Z) . x = - ((x ^2) / (1 + (x ^2)))
then ((arctan - (id Z)) `| Z) . x = (diff (arctan,x)) - (diff ((id Z),x)) by A1, A5, A6, FDIFF_1:19
.= ((arctan `| Z) . x) - (diff ((id Z),x)) by A6, A8, FDIFF_1:def 7
.= (1 / (1 + (x ^2))) - (diff ((id Z),x)) by A2, A8, Th81
.= (1 / (1 + (x ^2))) - (((id Z) `| Z) . x) by A5, A8, FDIFF_1:def 7
.= (1 / (1 + (x ^2))) - 1 by A4, A3, A8, FDIFF_1:23
.= (1 / (1 + (x ^2))) - ((1 + (x ^2)) / (1 + (x ^2))) by A7, XCMPLX_1:60
.= - ((x ^2) / (1 + (x ^2))) ;
hence ((arctan - (id Z)) `| Z) . x = - ((x ^2) / (1 + (x ^2))) ; :: thesis: verum

end;

hence ( arctan - (id Z) is_differentiable_on Z & ( for x being Real st x in Z holds
((arctan - (id Z)) `| Z) . x = - ((x ^2) / (1 + (x ^2))) ) ) by A1, A5, A6, FDIFF_1:19; :: thesis: verum