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= dom ((1/2)(#)((arccot)*f)) & (for x st x in Z holds f.x=2*x & f.x
> -1 & f.x < 1) implies (1/2)(#)((arccot)*f) is_differentiable_on Z & for x st
  x in Z holds (((1/2)(#)((arccot)*f))`|Z).x = -1/(1+(2*x)^2)
proof
  assume that
A1: Z c= dom ((1/2)(#)((arccot)*f)) and
A2: for x st x in Z holds f.x=2*x & f.x > -1 & f.x < 1;
A3: for x st x in Z holds f.x=2*x+0 & f.x > -1 & f.x < 1 by A2;
A4: Z c= dom ((arccot)*f) by A1,VALUED_1:def 5;
  then
A5: (arccot)*f is_differentiable_on Z by A3,Th88;
  for x st x in Z holds (((1/2)(#)((arccot)*f))`|Z).x = -1/(1+(2*x)^2)
  proof
    let x;
    assume
A6: x in Z;
    then (((1/2)(#)((arccot)*f))`|Z).x = (1/2)*diff(((arccot)*f),x) by A1,A5,
FDIFF_1:20
      .= (1/2)*(((arccot)*f)`|Z).x by A5,A6,FDIFF_1:def 7
      .= (1/2)*(-2/(1+(2*x+0)^2)) by A4,A3,A6,Th88
      .= -1/(1+(2*x)^2);
    hence thesis;
  end;
  hence thesis by A1,A5,FDIFF_1:20;
end;
