let Z be open Subset of REAL; :: thesis: ( not 0 in Z & Z c= dom (((id Z) ^) (#) arctan) & Z c= ].(- 1),1.[ implies ( ((id Z) ^) (#) arctan is_differentiable_on Z & ( for x being Real st x in Z holds
((((id Z) ^) (#) arctan) `| Z) . x = (- ((arctan . x) / (x ^2))) + (1 / (x * (1 + (x ^2)))) ) ) )
set f = id Z;
assume that
A1: not 0 in Z and
A2: Z c= dom (((id Z) ^) (#) arctan) and
A3: Z c= ].(- 1),1.[ ; :: thesis: ( ((id Z) ^) (#) arctan is_differentiable_on Z & ( for x being Real st x in Z holds
((((id Z) ^) (#) arctan) `| Z) . x = (- ((arctan . x) / (x ^2))) + (1 / (x * (1 + (x ^2)))) ) )
A4: (id Z) ^ is_differentiable_on Z by A1, FDIFF_5:4;
A5: arctan is_differentiable_on Z by A3, Th81;
Z c= (dom ((id Z) ^)) /\ (dom arctan) by A2, VALUED_1:def 4;
then A6: Z c= dom ((id Z) ^) by XBOOLE_1:18;
for x being Real st x in Z holds
((((id Z) ^) (#) arctan) `| Z) . x = (- ((arctan . x) / (x ^2))) + (1 / (x * (1 + (x ^2))))

proof

let x be Real; :: thesis: ( x in Z implies ((((id Z) ^) (#) arctan) `| Z) . x = (- ((arctan . x) / (x ^2))) + (1 / (x * (1 + (x ^2)))) )
assume A7: x in Z ; :: thesis: ((((id Z) ^) (#) arctan) `| Z) . x = (- ((arctan . x) / (x ^2))) + (1 / (x * (1 + (x ^2))))
then ((((id Z) ^) (#) arctan) `| Z) . x = ((arctan . x) * (diff (((id Z) ^),x))) + ((((id Z) ^) . x) * (diff (arctan,x))) by A2, A4, A5, FDIFF_1:21
.= ((arctan . x) * ((((id Z) ^) `| Z) . x)) + ((((id Z) ^) . x) * (diff (arctan,x))) by A4, A7, FDIFF_1:def 7
.= ((arctan . x) * (- (1 / (x ^2)))) + ((((id Z) ^) . x) * (diff (arctan,x))) by A1, A7, FDIFF_5:4
.= (- ((arctan . x) * (1 / (x ^2)))) + ((((id Z) ^) . x) * ((arctan `| Z) . x)) by A5, A7, FDIFF_1:def 7
.= (- (((arctan . x) * 1) / (x ^2))) + ((((id Z) ^) . x) * (1 / (1 + (x ^2)))) by A3, A7, Th81
.= (- ((arctan . x) / (x ^2))) + ((((id Z) . x) ") * (1 / (1 + (x ^2)))) by A6, A7, RFUNCT_1:def 2
.= (- ((arctan . x) / (x ^2))) + ((1 / x) * (1 / (1 + (x ^2)))) by A7, FUNCT_1:18
.= (- ((arctan . x) / (x ^2))) + ((1 * 1) / (x * (1 + (x ^2)))) by XCMPLX_1:76
.= (- ((arctan . x) / (x ^2))) + (1 / (x * (1 + (x ^2)))) ;
hence ((((id Z) ^) (#) arctan) `| Z) . x = (- ((arctan . x) / (x ^2))) + (1 / (x * (1 + (x ^2)))) ; :: thesis: verum

end;

hence ( ((id Z) ^) (#) arctan is_differentiable_on Z & ( for x being Real st x in Z holds
((((id Z) ^) (#) arctan) `| Z) . x = (- ((arctan . x) / (x ^2))) + (1 / (x * (1 + (x ^2)))) ) ) by A2, A4, A5, FDIFF_1:21; :: thesis: verum