reserve x for Real,

  n for Element of NAT,
   y for set,
  Z for open Subset of REAL,

     g for PartFunc of REAL,REAL;

theorem Th37:
  Z c= ].-1,1.[ implies (arctan+arccot) is_differentiable_on Z &
  for x st x in Z holds ((arctan+arccot)`|Z).x = 0
proof
  assume
A1: Z c= ].-1,1.[;
  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 st x in Z holds ((arctan+arccot)`|Z).x = 0
  proof
    let x;
    assume
A7: x in Z;
    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 thesis;
  end;
  hence thesis by A6,A2,A5,FDIFF_1:18;
end;
