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 ))) ) )
Z c= (dom arctan ) /\ (dom (id Z)) by A1, VALUED_1:12;
then A3: ( Z c= dom arctan & Z c= dom (id Z) ) by XBOOLE_1:18;
A4: for x being Real st x in Z holds
(id Z) . x = (1 * x) + 0 by FUNCT_1:35;
then A5: ( id Z is_differentiable_on Z & ( for x being Real st x in Z holds
((id Z) `| Z) . x = 1 ) ) by A3, FDIFF_1:31;
A6: arctan is_differentiable_on Z by A2, Th79;
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 ))) )
assume A7: x in Z ; :: thesis: ((arctan - (id Z)) `| Z) . x = - ((x ^2 ) / (1 + (x ^2 )))
A8: 1 + (x ^2 ) > 0 by XREAL_1:36, XREAL_1:65;
((arctan - (id Z)) `| Z) . x = (diff arctan ,x) - (diff (id Z),x) by A1, A5, A6, A7, FDIFF_1:27
.= ((arctan `| Z) . x) - (diff (id Z),x) by A6, A7, FDIFF_1:def 8
.= (1 / (1 + (x ^2 ))) - (diff (id Z),x) by A2, A7, Th79
.= (1 / (1 + (x ^2 ))) - (((id Z) `| Z) . x) by A5, A7, FDIFF_1:def 8
.= (1 / (1 + (x ^2 ))) - 1 by A3, A4, A7, FDIFF_1:31
.= (1 / (1 + (x ^2 ))) - ((1 + (x ^2 )) / (1 + (x ^2 ))) by A8, 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_1:27; :: thesis: verum