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
  Z c= dom (cos*(arctan-arccot)) & Z c= ].-1,1.[ implies cos*(arctan-
arccot) is_differentiable_on Z & for x st x in Z holds (cos*(arctan-arccot)`|Z)
  .x = -2*sin.(arctan.x-arccot.x)/(1+x^2)
proof
  assume that
A1: Z c= dom (cos*(arctan-arccot)) and
A2: Z c= ].-1,1.[;
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 A2,XBOOLE_1:1;
  ].-1,1.[ c= dom arctan by A3,SIN_COS9:23,XBOOLE_1:1;
  then Z c= dom arctan by A2,XBOOLE_1:1;
  then Z c= dom arctan /\ dom arccot by A4,XBOOLE_1:19;
  then
A5: Z c= dom (arctan-arccot) by VALUED_1:12;
A6: arctan-arccot is_differentiable_on Z by A2,Th38;
A7: for x st x in Z holds cos*(arctan-arccot) is_differentiable_in x
  proof
    let x;
    assume x in Z;
    then
A8: (arctan-arccot) is_differentiable_in x by A6,FDIFF_1:9;
    cos is_differentiable_in (arctan-arccot).x by SIN_COS:63;
    hence thesis by A8,FDIFF_2:13;
  end;
  then
A9: cos*(arctan-arccot) is_differentiable_on Z by A1,FDIFF_1:9;
  for x st x in Z holds (cos*(arctan-arccot)`|Z).x = -2*sin.(arctan.x-
  arccot.x)/(1+x^2)
  proof
    let x;
A10: cos is_differentiable_in (arctan-arccot).x by SIN_COS:63;
    assume
A11: x in Z;
    then
A12: arctan-arccot is_differentiable_in x by A6,FDIFF_1:9;
    (cos*(arctan-arccot)`|Z).x = diff(cos*(arctan-arccot),x) by A9,A11,
FDIFF_1:def 7
      .= diff(cos,(arctan-arccot).x)*diff((arctan-arccot),x) by A12,A10,
FDIFF_2:13
      .= (-sin.((arctan-arccot).x))*diff((arctan-arccot),x) by SIN_COS:63
      .= (-sin.((arctan-arccot).x))*((arctan-arccot)`|Z).x by A6,A11,
FDIFF_1:def 7
      .= (-sin.((arctan-arccot).x))*(2/(1+x^2)) by A2,A11,Th38
      .= (-sin.(arctan.x-arccot.x))*(2/(1+x^2)) by A5,A11,VALUED_1:13
      .= -2*sin.(arctan.x-arccot.x)/(1+x^2);
    hence thesis;
  end;
  hence thesis by A1,A7,FDIFF_1:9;
end;
