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 Th71:
  arctan is_differentiable_on tan.:].-PI/2,PI/2.[
proof
  set f = tan|].-PI/2,PI/2.[;
A1: dom(f") = rng (tan|].-PI/2,PI/2.[) by FUNCT_1:33
    .= tan.:].-PI/2,PI/2.[ by RELAT_1:115;
  dom f = dom tan /\ ].-PI/2,PI/2.[ by RELAT_1:61;
  then
A2: ].-PI/2,PI/2.[ c= dom f by Th1,XBOOLE_1:19;
A3: f is_differentiable_on ].-PI/2,PI/2.[ by Lm1,FDIFF_2:16;
A4: now
A5: for x0 st x0 in ].-PI/2,PI/2.[ holds 1/(cos.x0)^2 > 0
    proof
      let x0;
      assume x0 in ].-PI/2,PI/2.[;
      then 0 < cos.x0 by COMPTRIG:11;
      then (cos.x0)^2 > 0;
      then 1/(cos.x0)^2 > 0 /(cos.x0)^2;
      hence thesis;
    end;
    let x0 such that
A6: x0 in ].-PI/2,PI/2.[;
    diff(f,x0) = (f`|].-PI/2,PI/2.[).x0 by A3,A6,FDIFF_1:def 7
      .= (tan`|].-PI/2,PI/2.[).x0 by Lm1,FDIFF_2:16
      .= diff(tan,x0) by A6,Lm1,FDIFF_1:def 7
      .= 1/(cos.x0)^2 by A6,Lm3;
    hence 0 < diff(f,x0) by A6,A5;
  end;
  f|].-PI/2,PI/2.[ = f by RELAT_1:72;
  hence thesis by A1,A3,A2,A4,FDIFF_2:48;
end;
