let Z be open Subset of REAL; :: thesis: ( Z c= ].(- 1),1.[ implies ( arctan - arccot is_differentiable_on Z & ( for x being Real st x in Z holds
((arctan - arccot) `| Z) . x = 2 / (1 + (x ^2)) ) ) )
assume A1: Z c= ].(- 1),1.[ ; :: thesis: ( arctan - arccot is_differentiable_on Z & ( for x being Real st x in Z holds
((arctan - arccot) `| Z) . x = 2 / (1 + (x ^2)) ) )
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_1:12;
for x being Real st x in Z holds
((arctan - arccot) `| Z) . x = 2 / (1 + (x ^2))

proof

let x be Real; :: thesis: ( x in Z implies ((arctan - arccot) `| Z) . x = 2 / (1 + (x ^2)) )
assume A7: x in Z ; :: thesis: ((arctan - arccot) `| Z) . x = 2 / (1 + (x ^2))
then ((arctan - arccot) `| Z) . x = (diff (arctan,x)) - (diff (arccot,x)) by A6, A2, A5, FDIFF_1:19
.= ((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
.= 2 / (1 + (x ^2)) ;
hence ((arctan - arccot) `| Z) . x = 2 / (1 + (x ^2)) ; :: thesis: verum

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 A6, A2, A5, FDIFF_1:19; :: thesis: verum