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 1;
for x being Real st x in Z holds
((arctan + arccot) `| Z) . x = 0

let x be Real; :: thesis:
assume A7: x in Z ; :: thesis:
then ((arctan + arccot) `| Z) . x = (diff (arctan,x)) + (diff (arccot,x)) by A6, A2, A5, FDIFF_1:18
.= ((arctan `| Z) . x) + (diff (arccot,x)) by A2, A7, FDIFF_1:def 7
.= (1 / (1 + (x ^2))) + (diff (arccot,x)) by A1, A7, SIN_COS9:81
.= (1 / (1 + (x ^2))) + ((arccot `| Z) . x) by A5, A7, FDIFF_1:def 7
.= (1 / (1 + (x ^2))) + (- (1 / (1 + (x ^2)))) by A1, A7, SIN_COS9:82
.= 0 ;
hence ((arctan + arccot) `| Z) . x = 0 ; :: thesis:

end;

hence ( arctan + arccot is_differentiable_on Z & ( for x being Real st x in Z holds
((arctan + arccot) `| Z) . x = 0 ) ) by A6, A2, A5, FDIFF_1:18; :: thesis: