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
  not 0 in Z & Z c= dom((arctan*((id Z)^))(#)(arccot*((id Z)^))) & (for
x st x in Z holds ((id Z)^).x > -1 & ((id Z)^).x < 1) implies (arctan*((id Z)^)
)(#)(arccot*((id Z)^)) is_differentiable_on Z & for x st x in Z holds (((arctan
  *((id Z)^))(#)(arccot*((id Z)^)))`|Z).x = (arctan.(1/x)-arccot.(1/x))/(1+x^2)
proof
  set f = id Z;
  assume that
A1: not 0 in Z and
A2: Z c= dom((arctan*(f^))(#)(arccot*(f^))) and
A3: for x st x in Z holds (f^).x > -1 & (f^).x < 1;
A4: Z c= dom(arctan*(f^)) /\ dom(arccot*(f^)) by A2,VALUED_1:def 4;
  then
A5: Z c= dom (arctan*(f^)) by XBOOLE_1:18;
  then
A6: arctan*(f^) is_differentiable_on Z by A1,A3,SIN_COS9:111;
A7: Z c= dom (arccot*(f^)) by A4,XBOOLE_1:18;
  then
A8: arccot*(f^) is_differentiable_on Z by A1,A3,SIN_COS9:112;
  for y being object st y in Z holds y in dom (f^) by A5,FUNCT_1:11;
  then
A9: Z c= dom (f^) by TARSKI:def 3;
  for x st x in Z holds (((arctan*(f^))(#)(arccot*(f^)))`|Z).x = (arctan.
  (1/x)-arccot.(1/x))/(1+x^2)
  proof
    let x;
    assume
A10: x in Z;
    then (((arctan*(f^))(#)(arccot*(f^)))`|Z).x = ((arccot*(f^)).x)*diff((
    arctan*(f^)),x)+ ((arctan*(f^)).x)*diff(arccot*(f^),x) by A2,A6,A8,
FDIFF_1:21
      .= ((arccot*(f^)).x)*((arctan*(f^))`|Z).x+ ((arctan*(f^)).x)*diff(
    arccot*(f^),x) by A6,A10,FDIFF_1:def 7
      .= ((arccot*(f^)).x)*(-1/(1+x^2))+((arctan*(f^)).x)*diff(arccot*(f^),x
    ) by A1,A3,A5,A10,SIN_COS9:111
      .= ((arccot*(f^)).x)*(-1/(1+x^2))+((arctan*(f^)).x)*((arccot*(f^))`|Z)
    .x by A8,A10,FDIFF_1:def 7
      .= ((arccot*(f^)).x)*(-1/(1+x^2))+((arctan*(f^)).x)*(1/(1+x^2)) by A1,A3
,A7,A10,SIN_COS9:112
      .= (arccot.((f^).x))*(-1/(1+x^2))+((arctan*(f^)).x)*(1/(1+x^2)) by A7,A10
,FUNCT_1:12
      .= (arccot.((f.x)"))*(-1/(1+x^2))+((arctan*(f^)).x)*(1/(1+x^2)) by A9,A10
,RFUNCT_1:def 2
      .= (arccot.(1/x))*(-1/(1+x^2))+((arctan*(f^)).x)*(1/(1+x^2)) by A10,
FUNCT_1:18
      .= (arccot.(1/x))*(-1/(1+x^2))+(arctan.((f^).x))*(1/(1+x^2)) by A5,A10,
FUNCT_1:12
      .= (arccot.(1/x))*(-1/(1+x^2))+(arctan.((f.x)"))*(1/(1+x^2)) by A9,A10,
RFUNCT_1:def 2
      .= -(arccot.(1/x))*(1/(1+x^2))+(arctan.(1/x))*(1/(1+x^2)) by A10,
FUNCT_1:18
      .= (arctan.(1/x)-arccot.(1/x))/(1+x^2);
    hence thesis;
  end;
  hence thesis by A2,A6,A8,FDIFF_1:21;
end;
