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 Th55:
  rng(arctan | [.-1,1.]) = [.-PI/4,PI/4.]
proof
  now
    let y be object;
    thus y in [.-PI/4,PI/4.] implies
ex x be object st x in dom (arctan | [.-1,1
    .]) & y = (arctan | [.-1,1.]).x
    proof
      assume
A1:   y in [.-PI/4,PI/4.];
      then reconsider y1=y as Real;
      y1 in [.arctan.(-1),arctan.1.] \/ [.arctan.1,arctan.(-1).] by A1,Th37
,Th39,XBOOLE_0:def 3;
      then consider x be Real such that
A2:   x in [.-1,1.] and
A3:   y1 = arctan.x by Th23,Th53,FCONT_2:15;
      take x;
      thus thesis by A2,A3,Th23,FUNCT_1:49,RELAT_1:62;
    end;
    thus (ex x be object
st x in dom (arctan | [.-1,1.]) & y = (arctan | [.-1,1.]
    ).x) implies y in [.-PI/4,PI/4.]
    proof
      given x be object such that
A4:   x in dom (arctan | [.-1,1.]) and
A5:   y = (arctan | [.-1,1.]).x;
A6:   dom (arctan | [.-1,1.]) = [.-1,1.] by Th23,RELAT_1:62;
      then y = arctan.x by A4,A5,FUNCT_1:49;
      hence thesis by A4,A6,Th49;
    end;
  end;
  hence thesis by FUNCT_1:def 3;
end;
