let Z be open Subset of REAL ; :: thesis:
set f = id Z;
assume that
A1: ( not 0 in Z & Z c= dom ((arctan * ((id Z) ^ )) (#) (arccot * ((id Z) ^ ))) ) and
A2: for x being Real st x in Z holds
( ((id Z) ^ ) . x > - 1 & ((id Z) ^ ) . x < 1 ) ; :: thesis:
Z c= (dom (arctan * ((id Z) ^ ))) /\ (dom (arccot * ((id Z) ^ ))) by A1, VALUED_1:def 4;
then A3: ( Z c= dom (arctan * ((id Z) ^ )) & Z c= dom (arccot * ((id Z) ^ )) ) by XBOOLE_1:18;
then for y being set st y in Z holds
y in dom ((id Z) ^ ) by FUNCT_1:21;
then A4: Z c= dom ((id Z) ^ ) by TARSKI:def 3;
A5: ( arctan * ((id Z) ^ ) is_differentiable_on Z & ( for x being Real st x in Z holds
((arctan * ((id Z) ^ )) `| Z) . x = - (1 / (1 + (x ^2 ))) ) ) by A2, A3, SIN_COS9:111, A1;
A6: ( arccot * ((id Z) ^ ) is_differentiable_on Z & ( for x being Real st x in Z holds
((arccot * ((id Z) ^ )) `| Z) . x = 1 / (1 + (x ^2 )) ) ) by A2, A3, SIN_COS9:112, A1;
for x being Real st x in Z holds
(((arctan * ((id Z) ^ )) (#) (arccot * ((id Z) ^ ))) `| Z) . x = ((arctan . (1 / x)) - (arccot . (1 / x))) / (1 + (x ^2 ))

proof

let x be Real; :: thesis:
assume A8: x in Z ; :: thesis:
(((arctan * ((id Z) ^ )) (#) (arccot * ((id Z) ^ ))) `| Z) . x = (((arccot * ((id Z) ^ )) . x) * (diff (arctan * ((id Z) ^ )),x)) + (((arctan * ((id Z) ^ )) . x) * (diff (arccot * ((id Z) ^ )),x)) by A1, A5, A6, A8, FDIFF_1:29
.= (((arccot * ((id Z) ^ )) . x) * (((arctan * ((id Z) ^ )) `| Z) . x)) + (((arctan * ((id Z) ^ )) . x) * (diff (arccot * ((id Z) ^ )),x)) by A5, A8, FDIFF_1:def 8
.= (((arccot * ((id Z) ^ )) . x) * (- (1 / (1 + (x ^2 ))))) + (((arctan * ((id Z) ^ )) . x) * (diff (arccot * ((id Z) ^ )),x)) by A2, A3, A8, SIN_COS9:111, A1
.= (((arccot * ((id Z) ^ )) . x) * (- (1 / (1 + (x ^2 ))))) + (((arctan * ((id Z) ^ )) . x) * (((arccot * ((id Z) ^ )) `| Z) . x)) by A6, A8, FDIFF_1:def 8
.= (((arccot * ((id Z) ^ )) . x) * (- (1 / (1 + (x ^2 ))))) + (((arctan * ((id Z) ^ )) . x) * (1 / (1 + (x ^2 )))) by A2, A3, A8, SIN_COS9:112, A1
.= ((arccot . (((id Z) ^ ) . x)) * (- (1 / (1 + (x ^2 ))))) + (((arctan * ((id Z) ^ )) . x) * (1 / (1 + (x ^2 )))) by A3, A8, FUNCT_1:22
.= ((arccot . (((id Z) . x) " )) * (- (1 / (1 + (x ^2 ))))) + (((arctan * ((id Z) ^ )) . x) * (1 / (1 + (x ^2 )))) by A4, A8, RFUNCT_1:def 8
.= ((arccot . (1 / x)) * (- (1 / (1 + (x ^2 ))))) + (((arctan * ((id Z) ^ )) . x) * (1 / (1 + (x ^2 )))) by A8, FUNCT_1:35
.= ((arccot . (1 / x)) * (- (1 / (1 + (x ^2 ))))) + ((arctan . (((id Z) ^ ) . x)) * (1 / (1 + (x ^2 )))) by A3, A8, FUNCT_1:22
.= ((arccot . (1 / x)) * (- (1 / (1 + (x ^2 ))))) + ((arctan . (((id Z) . x) " )) * (1 / (1 + (x ^2 )))) by A4, A8, RFUNCT_1:def 8
.= (- ((arccot . (1 / x)) * (1 / (1 + (x ^2 ))))) + ((arctan . (1 / x)) * (1 / (1 + (x ^2 )))) by A8, FUNCT_1:35
.= ((arctan . (1 / x)) - (arccot . (1 / x))) / (1 + (x ^2 )) ;
hence (((arctan * ((id Z) ^ )) (#) (arccot * ((id Z) ^ ))) `| Z) . x = ((arctan . (1 / x)) - (arccot . (1 / x))) / (1 + (x ^2 )) ; :: thesis:

end;

hence ( (arctan * ((id Z) ^ )) (#) (arccot * ((id Z) ^ )) is_differentiable_on Z & ( for x being Real st x in Z holds
(((arctan * ((id Z) ^ )) (#) (arccot * ((id Z) ^ ))) `| Z) . x = ((arctan . (1 / x)) - (arccot . (1 / x))) / (1 + (x ^2 )) ) ) by A1, A5, A6, FDIFF_1:29; :: thesis: