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 (arctan-arccot)/exp_R is_differentiable_on Z &
  for x st x in Z holds (((arctan-arccot)/exp_R)`|Z).x = ((2/(1+x^2))-arctan.x+
  arccot.x)/exp_R.x
proof
  assume
A1: Z c= ].-1,1.[;
  then
A2: arctan-arccot is_differentiable_on Z by Th38;
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_1:12;
A6: exp_R is_differentiable_on Z & for x st x in Z holds exp_R.x<>0 by
FDIFF_1:26,SIN_COS:54,TAYLOR_1:16;
  then
A7: (arctan-arccot)/exp_R is_differentiable_on Z by A2,FDIFF_2:21;
  for x st x in Z holds (((arctan-arccot)/exp_R)`|Z).x = ((2/(1+x^2))-
  arctan.x+arccot.x)/exp_R.x
  proof
    let x;
A8: exp_R is_differentiable_in x by SIN_COS:65;
A9: exp_R.x <>0 by SIN_COS:54;
    assume
A10: x in Z;
    then
A11: arctan-arccot is_differentiable_in x by A2,FDIFF_1:9;
    (((arctan-arccot)/exp_R)`|Z).x = diff((arctan-arccot)/exp_R,x) by A7,A10,
FDIFF_1:def 7
      .= (diff(arctan-arccot,x)*exp_R.x-diff(exp_R,x)*(arctan-arccot).x) /(
    exp_R.x)^2 by A11,A8,A9,FDIFF_2:14
      .= (((arctan-arccot)`|Z).x*exp_R.x-diff(exp_R,x)*(arctan-arccot).x) /(
    exp_R.x)^2 by A2,A10,FDIFF_1:def 7
      .= ((2/(1+x^2))*exp_R.x-diff(exp_R,x)*(arctan-arccot).x)/(exp_R.x)^2
    by A1,A10,Th38
      .= ((2/(1+x^2))*exp_R.x-exp_R.x*(arctan-arccot).x)/(exp_R.x)^2 by
SIN_COS:65
      .= ((2/(1+x^2))*exp_R.x-exp_R.x*(arctan.x-arccot.x))/(exp_R.x)^2 by A5
,A10,VALUED_1:13
      .= ((2/(1+x^2))-(arctan.x-arccot.x))*(exp_R.x/(exp_R.x*exp_R.x))
      .= ((2/(1+x^2))-(arctan.x-arccot.x))*(exp_R.x/exp_R.x/exp_R.x) by
XCMPLX_1:78
      .= ((2/(1+x^2))-(arctan.x-arccot.x))*(1/exp_R.x) by A9,XCMPLX_1:60
      .= ((2/(1+x^2))-arctan.x+arccot.x)/exp_R.x;
    hence thesis;
  end;
  hence thesis by A2,A6,FDIFF_2:21;
end;
