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 ((1/n)(#)(( #Z n)*(arctan^))) & Z c= ].-1,1.[ & n>0 & (for x
  st x in Z holds arctan.x<>0) implies (1/n)(#)(( #Z n)*(arctan^))
is_differentiable_on Z & for x st x in Z holds (((1/n)(#)(( #Z n)*(arctan^)))`|
  Z).x = -1/(((arctan.x) #Z (n+1))*(1+x^2))
proof
  assume that
A1: Z c= dom ((1/n)(#)(( #Z n)*(arctan^))) and
A2: Z c= ].-1,1.[ and
A3: n>0 and
A4: for x st x in Z holds arctan.x<>0;
A5: Z c= dom (( #Z n)*(arctan^)) by A1,VALUED_1:def 5;
A6: arctan^ is_differentiable_on Z by A2,A4,Th67;
  for x st x in Z holds ( #Z n)*(arctan^) is_differentiable_in x
  proof
    let x;
    assume x in Z;
    then arctan^ is_differentiable_in x by A6,FDIFF_1:9;
    hence thesis by TAYLOR_1:3;
  end;
  then
A7: ( #Z n)*(arctan^) is_differentiable_on Z by A5,FDIFF_1:9;
  for y being object st y in Z holds y in dom (arctan^) by A5,FUNCT_1:11;
  then
A8: Z c= dom (arctan^) by TARSKI:def 3;
  for x st x in Z holds (((1/n)(#)(( #Z n)*(arctan^)))`|Z).x = -1/(((
  arctan.x) #Z (n+1))*(1+x^2))
  proof
    let x;
    assume
A9: x in Z;
    then
A10: arctan^ is_differentiable_in x by A6,FDIFF_1:9;
A11: (arctan^).x = 1/arctan.x by A8,A9,RFUNCT_1:def 2;
    (((1/n)(#)(( #Z n)*(arctan^)))`|Z).x = (1/n)*diff((( #Z n)*(arctan^))
    ,x) by A1,A7,A9,FDIFF_1:20
      .= (1/n)*(n*(((arctan^).x) #Z (n-1))*diff(arctan^,x)) by A10,TAYLOR_1:3
      .= (1/n)*(n*(((arctan^).x) #Z (n-1))*((arctan^)`|Z).x) by A6,A9,
FDIFF_1:def 7
      .= (1/n)*(n*(((arctan^).x) #Z (n-1))*(-1/((arctan.x)^2*(1+x^2)))) by A2
,A4,A9,Th67
      .= -((1/n)*n)*(((arctan^).x) #Z (n-1))*(1/((arctan.x)^2*(1+x^2)))
      .= -1*(((arctan^).x) #Z (n-1))*(1/((arctan.x)^2*(1+x^2))) by A3,
XCMPLX_1:106
      .= -((1/arctan.x) #Z (n-1))*(1/(((arctan.x) #Z 2)*(1+x^2))) by A11,
FDIFF_7:1
      .= -(1/((arctan.x) #Z (n-1)))/(((arctan.x) #Z 2)*(1+x^2)) by PREPOWER:42
      .= -1/(((arctan.x) #Z (n-1))*(((arctan.x) #Z 2)*(1+x^2))) by XCMPLX_1:78
      .= -1/(((arctan.x) #Z (n-1))*((arctan.x) #Z 2)*(1+x^2))
      .= -1/(((arctan.x) #Z ((n-1)+2))*(1+x^2)) by A4,A9,PREPOWER:44
      .= -1/(((arctan.x) #Z (n+1))*(1+x^2));
    hence thesis;
  end;
  hence thesis by A1,A7,FDIFF_1:20;
end;
