let Z be open Subset of REAL; :: thesis:
assume A1: Z c= ].(- 1),1.[ ; :: thesis:
then A2: arctan is_differentiable_on Z by SIN_COS9:81;
A3: ].(- 1),1.[ c= [.(- 1),1.] by XXREAL_1:25;
then ].(- 1),1.[ c= dom arccot by SIN_COS9:24, XBOOLE_1:1;
then A4: Z c= dom arccot by A1, XBOOLE_1:1;
A5: arccot is_differentiable_on Z by A1, SIN_COS9:82;
].(- 1),1.[ c= dom arctan by A3, SIN_COS9:23, XBOOLE_1:1;
then Z c= dom arctan by A1, XBOOLE_1:1;
then Z c= (dom arctan) /\ (dom arccot) by A4, XBOOLE_1:19;
then A6: Z c= dom (arctan (#) arccot) by VALUED_1:def 4;
for x being Real st x in Z holds
((arctan (#) arccot) `| Z) . x = ((arccot . x) - (arctan . x)) / (1 + (x ^2))

proof

let x be Real; :: thesis:
assume A7: x in Z ; :: thesis:
then ((arctan (#) arccot) `| Z) . x = ((arccot . x) * (diff (arctan,x))) + ((arctan . x) * (diff (arccot,x))) by A6, A2, A5, FDIFF_1:21
.= ((arccot . x) * ((arctan `| Z) . x)) + ((arctan . x) * (diff (arccot,x))) by A2, A7, FDIFF_1:def 7
.= ((arccot . x) * (1 / (1 + (x ^2)))) + ((arctan . x) * (diff (arccot,x))) by A1, A7, SIN_COS9:81
.= ((arccot . x) * (1 / (1 + (x ^2)))) + ((arctan . x) * ((arccot `| Z) . x)) by A5, A7, FDIFF_1:def 7
.= ((arccot . x) * (1 / (1 + (x ^2)))) + ((arctan . x) * (- (1 / (1 + (x ^2))))) by A1, A7, SIN_COS9:82
.= ((arccot . x) - (arctan . x)) / (1 + (x ^2)) ;
hence ((arctan (#) arccot) `| Z) . x = ((arccot . x) - (arctan . x)) / (1 + (x ^2)) ; :: thesis:

end;

hence ( arctan (#) arccot is_differentiable_on Z & ( for x being Real st x in Z holds
((arctan (#) arccot) `| Z) . x = ((arccot . x) - (arctan . x)) / (1 + (x ^2)) ) ) by A6, A2, A5, FDIFF_1:21; :: thesis: