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= ].-1,1.[ implies exp_R(#)(arctan+arccot) is_differentiable_on Z &
  for x st x in Z holds ((exp_R(#)(arctan+arccot))`|Z).x = exp_R.x*(arctan.x+
  arccot.x)
proof
  assume
A1: Z c= ].-1,1.[;
  then
A2: arctan+arccot is_differentiable_on Z by Th37;
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;
  ].-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
A5: Z c= dom (arctan+arccot) by VALUED_1:def 1;
  then Z c= dom exp_R /\ dom (arctan+arccot) by TAYLOR_1:16,XBOOLE_1:19;
  then
A6: Z c= dom (exp_R(#)(arctan+arccot)) by VALUED_1:def 4;
A7: exp_R is_differentiable_on Z by FDIFF_1:26,TAYLOR_1:16;
  for x st x in Z holds (exp_R(#)(arctan+arccot)`|Z).x = exp_R.x*(arctan.
  x+arccot.x)
  proof
    let x;
    assume
A8: x in Z;
    then (exp_R(#)(arctan+arccot)`|Z).x = ((arctan+arccot).x)*diff(exp_R,x)+
    exp_R.x*diff(arctan+arccot,x) by A6,A7,A2,FDIFF_1:21
      .= (arctan.x+arccot.x)*diff(exp_R,x)+exp_R.x*diff(arctan+arccot,x) by A5
,A8,VALUED_1:def 1
      .= (arctan.x+arccot.x)*exp_R.x+exp_R.x*diff(arctan+arccot,x) by
TAYLOR_1:16
      .= (arctan.x+arccot.x)*exp_R.x+exp_R.x*((arctan+arccot)`|Z).x by A2,A8,
FDIFF_1:def 7
      .= (arctan.x+arccot.x)*exp_R.x+exp_R.x*0 by A1,A8,Th37
      .= exp_R.x*(arctan.x+arccot.x);
    hence thesis;
  end;
  hence thesis by A6,A7,A2,FDIFF_1:21;
end;
