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 tan & Z c= ].-1,1.[ implies tan(#)(arctan+arccot)
is_differentiable_on Z & for x st x in Z holds ((tan(#)(arctan+arccot))`|Z).x =
  (arctan.x+arccot.x)/(cos.x)^2
proof
  assume that
A1: Z c= dom tan and
A2: Z c= ].-1,1.[;
A3: arctan+arccot is_differentiable_on Z by A2,Th37;
  for x st x in Z holds tan is_differentiable_in x
  proof
    let x;
    assume x in Z;
    then cos.x <> 0 by A1,FDIFF_8:1;
    hence thesis by FDIFF_7:46;
  end;
  then
A4: tan is_differentiable_on Z by A1,FDIFF_1:9;
A5: ].-1,1.[ c= [.-1,1.] by XXREAL_1:25;
  then ].-1,1.[ c= dom arccot by SIN_COS9:24,XBOOLE_1:1;
  then
A6: Z c= dom arccot by A2,XBOOLE_1:1;
  ].-1,1.[ c= dom arctan by A5,SIN_COS9:23,XBOOLE_1:1;
  then Z c= dom arctan by A2,XBOOLE_1:1;
  then Z c= dom arctan /\ dom arccot by A6,XBOOLE_1:19;
  then
A7: Z c= dom (arctan+arccot) by VALUED_1:def 1;
  then Z c= dom tan /\ dom (arctan+arccot) by A1,XBOOLE_1:19;
  then
A8: Z c= dom (tan(#)(arctan+arccot)) by VALUED_1:def 4;
  for x st x in Z holds (tan(#)(arctan+arccot)`|Z).x = (arctan.x+arccot.x
  )/(cos.x)^2
  proof
    let x;
    assume
A9: x in Z;
    then
A10: cos.x <> 0 by A1,FDIFF_8:1;
    (tan(#)(arctan+arccot)`|Z).x = ((arctan+arccot).x)*diff(tan,x)+tan.x*
    diff(arctan+arccot,x) by A8,A4,A3,A9,FDIFF_1:21
      .= (arctan.x+arccot.x)*diff(tan,x)+tan.x*diff(arctan+arccot,x) by A7,A9,
VALUED_1:def 1
      .= (arctan.x+arccot.x)*(1/(cos.x)^2)+tan.x*diff(arctan+arccot,x) by A10,
FDIFF_7:46
      .= (arctan.x+arccot.x)/(cos.x)^2+tan.x*((arctan+arccot)`|Z).x by A3,A9,
FDIFF_1:def 7
      .= (arctan.x+arccot.x)/(cos.x)^2+tan.x*0 by A2,A9,Th37
      .= (arctan.x+arccot.x)/(cos.x)^2;
    hence thesis;
  end;
  hence thesis by A8,A4,A3,FDIFF_1:21;
end;
