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 = 0
proof
  assume that
A1: Z c= dom (cos*(arctan+arccot)) and
A2: Z c= ].-1,1.[;
A3: arctan+arccot is_differentiable_on Z by A2,Th37;
A4: for x st x in Z holds cos*(arctan+arccot) is_differentiable_in x
  proof
    let x;
    assume x in Z;
    then
A5: (arctan+arccot) is_differentiable_in x by A3,FDIFF_1:9;
    cos is_differentiable_in (arctan+arccot).x by SIN_COS:63;
    hence thesis by A5,FDIFF_2:13;
  end;
  then
A6: cos*(arctan+arccot) is_differentiable_on Z by A1,FDIFF_1:9;
  for x st x in Z holds (cos*(arctan+arccot)`|Z).x = 0
  proof
    let x;
A7: cos is_differentiable_in (arctan+arccot).x by SIN_COS:63;
    assume
A8: x in Z;
    then
A9: arctan+arccot is_differentiable_in x by A3,FDIFF_1:9;
    (cos*(arctan+arccot)`|Z).x = diff(cos*(arctan+arccot),x) by A6,A8,
FDIFF_1:def 7
      .= diff(cos,(arctan+arccot).x)*diff((arctan+arccot),x) by A9,A7,
FDIFF_2:13
      .= (-sin.((arctan+arccot).x))*diff((arctan+arccot),x) by SIN_COS:63
      .= (-sin.((arctan+arccot).x))*((arctan+arccot)`|Z).x by A3,A8,
FDIFF_1:def 7
      .= (-sin.((arctan+arccot).x))*0 by A2,A8,Th37
      .= 0;
    hence thesis;
  end;
  hence thesis by A1,A4,FDIFF_1:9;
end;
