let Z be open Subset of REAL ; :: thesis:
assume A1: Z c= ].(- 1),1.[ ; :: thesis:
A2: ].(- 1),1.[ c= [.(- 1),1.] by XXREAL_1:25;
then ].(- 1),1.[ c= dom arctan by SIN_COS9:23, XBOOLE_1:1;
then A3: Z c= dom arctan by A1, XBOOLE_1:1;
].(- 1),1.[ c= dom arccot by A2, SIN_COS9:24, XBOOLE_1:1;
then Z c= dom arccot by A1, XBOOLE_1:1;
then Z c= (dom arctan ) /\ (dom arccot ) by A3, XBOOLE_1:19;
then A4: Z c= dom (arctan - arccot ) by VALUED_1:12;
A5: arctan is_differentiable_on Z by A1, SIN_COS9:81;
A6: arccot is_differentiable_on Z by A1, SIN_COS9:82;
for x being Real st x in Z holds
((arctan - arccot ) `| Z) . x = 2 / (1 + (x ^2 ))

let x be Real; :: thesis:
assume A7: x in Z ; :: thesis:
((arctan - arccot ) `| Z) . x = (diff arctan ,x) - (diff arccot ,x) by A4, A5, A6, A7, FDIFF_1:27
.= ((arctan `| Z) . x) - (diff arccot ,x) by A5, A7, FDIFF_1:def 8
.= (1 / (1 + (x ^2 ))) - (diff arccot ,x) by A1, A7, SIN_COS9:81
.= (1 / (1 + (x ^2 ))) - ((arccot `| Z) . x) by A6, A7, FDIFF_1:def 8
.= (1 / (1 + (x ^2 ))) - (- (1 / (1 + (x ^2 )))) by A1, A7, SIN_COS9:82
.= 2 / (1 + (x ^2 )) ;
hence ((arctan - arccot ) `| Z) . x = 2 / (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 = 2 / (1 + (x ^2 )) ) ) by A4, A5, A6, FDIFF_1:27; :: thesis: