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 Th85:
  f is_differentiable_in x & f.x > -1 & f.x < 1 implies (arctan)*f
  is_differentiable_in x & diff((arctan)*f,x) = diff(f,x)/(1+(f.x)^2)
proof
  assume that
A1: f is_differentiable_in x and
A2: f.x > -1 and
A3: f.x < 1;
  f.x in ].-1,1.[ by A2,A3,XXREAL_1:4;
  then
A4: arctan is_differentiable_in f.x by Th73,FDIFF_1:9;
  then diff(arctan*f,x) = diff(arctan,f.x)*diff(f,x) by A1,FDIFF_2:13
    .= diff(f,x)*(1/(1+(f.x)^2)) by A2,A3,Th75
    .= diff(f,x)/(1+(f.x)^2);
  hence thesis by A1,A4,FDIFF_2:13;
end;
