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 & Z c= dom (((id Z) ^ ) (#) arctan ) ) and
A2: 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 )))) ) )
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;
A7: ( (id Z) ^ is_differentiable_on Z & ( for x being Real st x in Z holds
(((id Z) ^ ) `| Z) . x = - (1 / (x ^2 )) ) ) by A1, FDIFF_5:4;
A8: arctan is_differentiable_on Z by A2, Th79;
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 ((((id Z) ^ ) (#) arctan ) `| Z) . x = ((arctan . x) * (diff ((id Z) ^ ),x)) + ((((id Z) ^ ) . x) * (diff arctan ,x)) by A1, A7, A8, FDIFF_1:29
.= ((arctan . x) * ((((id Z) ^ ) `| Z) . x)) + ((((id Z) ^ ) . x) * (diff arctan ,x)) by A7, A9, FDIFF_1:def 8
.= ((arctan . x) * (- (1 / (x ^2 )))) + ((((id Z) ^ ) . x) * (diff arctan ,x)) by A9, A1, FDIFF_5:4
.= (- ((arctan . x) * (1 / (x ^2 )))) + ((((id Z) ^ ) . x) * ((arctan `| Z) . x)) by A8, A9, FDIFF_1:def 8
.= (- (((arctan . x) * 1) / (x ^2 ))) + ((((id Z) ^ ) . x) * (1 / (1 + (x ^2 )))) by A2, A9, Th79
.= (- ((arctan . x) / (x ^2 ))) + ((((id Z) . x) " ) * (1 / (1 + (x ^2 )))) by A4, A9, RFUNCT_1:def 8
.= (- ((arctan . x) / (x ^2 ))) + ((1 / x) * (1 / (1 + (x ^2 )))) by A3, A9
.= (- ((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 A1, A7, A8, FDIFF_1:29; :: thesis: verum