let Z be open Subset of REAL; :: thesis: ( 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: Z c= dom (((id Z) ^) (#) arctan) and
B: 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)))) ) )
A2: Z c= dom (- (((id Z) ^) (#) arctan)) by A1, VALUED_1:8;
A3: for x being Real st x in Z holds
(id Z) . x = x by FUNCT_1:35;
Z c= (dom ((id Z) ^)) /\ (dom arctan) by A1, VALUED_1:def 4;
then A4: Z c= dom ((id Z) ^) by XBOOLE_1:18;
A0: not 0 in Z

proof

assume K: 0 in Z ; :: thesis: contradiction
dom ((id Z) ^) = (dom (id Z)) \ ((id Z) " {0}) by RFUNCT_1:def 8
.= (dom (id Z)) \ {0} by Lm0, K ;
then not 0 in {0} by XBOOLE_0:def 5, K, A4;
hence contradiction by TARSKI:def 1; :: thesis: verum

end;

then A5: ( (id Z) ^ is_differentiable_on Z & ( for x being Real st x in Z holds
(((id Z) ^) `| Z) . x = - (1 / (x ^2)) ) ) by FDIFF_5:4;
A6: arctan is_differentiable_on Z by B, SIN_COS9:81;
A7: ((id Z) ^) (#) arctan is_differentiable_on Z by A1, B, A0, SIN_COS9:129;
then A8: (- 1) (#) (((id Z) ^) (#) arctan) is_differentiable_on Z by A2, FDIFF_1:28, A;
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 A9: x in Z ; :: thesis: ((- (((id Z) ^) (#) arctan)) `| Z) . x = ((arctan . x) / (x ^2)) - (1 / (x * (1 + (x ^2))))
then A10: ((id Z) ^) (#) arctan is_differentiable_in x by A7, FDIFF_1:16;
A11: (id Z) ^ is_differentiable_in x by A5, A9, FDIFF_1:16;
A12: arctan is_differentiable_in x by A6, A9, FDIFF_1:16;
((- (((id Z) ^) (#) arctan)) `| Z) . x = diff ((- (((id Z) ^) (#) arctan)),x) by A8, A9, FDIFF_1:def 8
.= (- 1) * (diff ((((id Z) ^) (#) arctan),x)) by A10, FDIFF_1:23, A
.= (- 1) * (((arctan . x) * (diff (((id Z) ^),x))) + ((((id Z) ^) . x) * (diff (arctan,x)))) by A11, A12, FDIFF_1:24
.= (- 1) * (((arctan . x) * ((((id Z) ^) `| Z) . x)) + ((((id Z) ^) . x) * (diff (arctan,x)))) by A5, A9, FDIFF_1:def 8
.= (- 1) * (((arctan . x) * (- (1 / (x ^2)))) + ((((id Z) ^) . x) * (diff (arctan,x)))) by A0, A9, FDIFF_5:4
.= (- 1) * ((- ((arctan . x) * (1 / (x ^2)))) + ((((id Z) ^) . x) * ((arctan `| Z) . x))) by A6, A9, FDIFF_1:def 8
.= (- 1) * ((- (((arctan . x) * 1) / (x ^2))) + ((((id Z) ^) . x) * (1 / (1 + (x ^2))))) by B, A9, SIN_COS9:81
.= (- 1) * ((- ((arctan . x) / (x ^2))) + ((((id Z) . x) ") * (1 / (1 + (x ^2))))) by A4, A9, RFUNCT_1:def 8
.= (- 1) * ((- ((arctan . x) / (x ^2))) + ((1 / x) * (1 / (1 + (x ^2))))) by A3, A9
.= (- 1) * ((- ((arctan . x) / (x ^2))) + ((1 * 1) / (x * (1 + (x ^2))))) by XCMPLX_1:77
.= ((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, A7, FDIFF_1:28, A; :: thesis: verum