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 ((2(#)(( #R (1/2))*arctan))) & Z c= ].-1,1.[ & (for x st x in
Z holds arctan.x > 0) implies 2(#)(( #R (1/2))*arctan) is_differentiable_on Z &
for x st x in Z holds ((2(#)(( #R (1/2))*arctan))`|Z).x = (arctan.x) #R (-1/2)/
  (1+x^2)
proof
  assume that
A1: Z c= dom ((2(#)(( #R (1/2))*arctan))) and
A2: Z c= ].-1,1.[ and
A3: for x st x in Z holds arctan.x > 0;
A4: for x st x in Z holds ( #R (1/2))*arctan is_differentiable_in x
  proof
    let x;
    assume
A5: x in Z;
    then
A6: arctan.x > 0 by A3;
    arctan is_differentiable_on Z by A2,SIN_COS9:81;
    then arctan is_differentiable_in x by A5,FDIFF_1:9;
    hence thesis by A6,TAYLOR_1:22;
  end;
  Z c= dom (( #R (1/2))*arctan) by A1,VALUED_1:def 5;
  then
A7: ( #R (1/2))*arctan is_differentiable_on Z by A4,FDIFF_1:9;
  for x st x in Z holds ((2(#)(( #R (1/2))*arctan))`|Z).x = ((arctan.x)
  #R (-1/2))/(1+x^2)
  proof
    let x;
    assume
A8: x in Z;
    then
A9: arctan.x > 0 by A3;
A10: arctan is_differentiable_on Z by A2,SIN_COS9:81;
    then
A11: arctan is_differentiable_in x by A8,FDIFF_1:9;
    ((2(#)(( #R (1/2))*arctan))`|Z).x = 2*diff((( #R (1/2))*arctan),x) by A1,A7
,A8,FDIFF_1:20
      .= 2*((1/2)*((arctan.x) #R (1/2-1))*diff(arctan,x)) by A11,A9,TAYLOR_1:22
      .= 2*((1/2)*((arctan.x) #R (1/2-1))*((arctan)`|Z).x) by A8,A10,
FDIFF_1:def 7
      .= 2*((1/2)*((arctan.x) #R (1/2-1))*(1/(1+x^2))) by A2,A8,SIN_COS9:81
      .= ((arctan.x) #R (-1/2))/(1+x^2);
    hence thesis;
  end;
  hence thesis by A1,A7,FDIFF_1:20;
end;
