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 Th23:
  [.-1,1.] c= dom arctan
proof
A1: [.-PI/4,PI/4.] c= ].-PI/2,PI/2.[ by Lm7,Lm8,XXREAL_2:def 12;
  rng (tan | [.-PI/4,PI/4.]) c= rng (tan | ].-PI/2,PI/2.[)
  proof
    let y be object;
    assume y in rng (tan | [.-PI/4,PI/4.]);
    then y in tan.:[.-PI/4,PI/4.] by RELAT_1:115;
    then ex x be object st x in dom tan & x in [.-PI/4,PI/4.] & y = tan.x by
FUNCT_1:def 6;
    then y in tan.:].-PI/2,PI/2.[ by A1,FUNCT_1:def 6;
    hence thesis by RELAT_1:115;
  end;
  hence thesis by Th21,FUNCT_1:33;
end;
