reserve x,x0, r, s, h for Real,

  n for Element of NAT,
  rr, y for set,
  Z for open Subset of REAL,

  f, f1, f2 for PartFunc of REAL,REAL;

theorem
  Z c= ].-1,1.[ implies (exp_R(#)arccot) is_differentiable_on Z & for x
  st x in Z holds ((exp_R(#)arccot)`|Z).x = exp_R.x*arccot.x-exp_R.x/(1+x^2)
proof
  assume
A1: Z c= ].-1,1.[;
  then
A2: arccot is_differentiable_on Z by Th82;
  ].-1,1.[ c= [.-1,1.] by XXREAL_1:25;
  then ].-1,1.[ c= dom arccot by Th24;
  then Z c= dom arccot by A1;
  then Z c= dom (exp_R) /\ dom arccot by SIN_COS:47,XBOOLE_1:19;
  then
A3: Z c= dom (exp_R(#)arccot) by VALUED_1:def 4;
  for x st x in Z holds exp_R is_differentiable_in x by SIN_COS:65;
  then
A4: exp_R is_differentiable_on Z by FDIFF_1:9,SIN_COS:47;
  for x st x in Z holds ((exp_R(#)arccot)`|Z).x = exp_R.x*arccot.x-exp_R.
  x/(1+x^2)
  proof
    let x;
    assume
A5: x in Z;
    then ((exp_R(#)arccot)`|Z).x = (arccot.x)*diff(exp_R,x)+(exp_R.x)*diff(
    arccot,x) by A3,A4,A2,FDIFF_1:21
      .= (arccot.x)*exp_R.x+(exp_R.x)*diff(arccot,x) by SIN_COS:65
      .= exp_R.x*arccot.x+(exp_R.x)*((arccot)`|Z).x by A2,A5,FDIFF_1:def 7
      .= exp_R.x*arccot.x+(exp_R.x)*(-1/(1+x^2)) by A1,A5,Th82
      .= exp_R.x*arccot.x-exp_R.x/(1+x^2);
    hence thesis;
  end;
  hence thesis by A3,A4,A2,FDIFF_1:21;
end;
