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 Th73:
  arctan is_differentiable_on ].-1,1.[
proof
  ].-1,1.[ c= [.-1,1.] by XXREAL_1:25;
  then
A1: ].-1,1.[ c= dom arctan by Th23;
  for x st x in ].-1,1.[ holds arctan is_differentiable_in x
  proof
    let x;
A2: dom arctan = rng (tan | ].-PI/2,PI/2.[) by FUNCT_1:33
      .= tan.:].-PI/2,PI/2.[ by RELAT_1:115;
    assume x in ].-1,1.[;
    then arctan|dom arctan is_differentiable_in x by A1,A2,Th71,FDIFF_1:def 6;
    hence thesis by RELAT_1:69;
  end;
  hence thesis by A1,FDIFF_1:9;
end;
