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_1:29
.= ((arctan . x) * ((((id Z) ^ ) `| Z) . x)) + ((((id Z) ^ ) . x) * (diff arctan ,x)) by A4, A7, FDIFF_1:def 8
.= ((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 8
.= (- (((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 8
.= (- ((arctan . x) / (x ^2 ))) + ((1 / x) * (1 / (1 + (x ^2 )))) by A7, FUNCT_1:35
.= (- ((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, A4, A5, FDIFF_1:29; :: thesis: verum