
theorem Th2:
  for f being one-to-one Function, y1, y2 being object st rng f = {y1, y2}
  holds dom f = {f".y1, f".y2}
proof
  let f be one-to-one Function, y1, y2 be object;
  assume A1: rng f = {y1,y2};
  then A2: y1 in rng f & y2 in rng f by TARSKI:def 2;
  then consider x1 being object such that
    A3: x1 in dom f & f.x1 = y1 by FUNCT_1:def 3;
  consider x2 being object such that
    A4: x2 in dom f & f.x2 = y2 by A2, FUNCT_1:def 3;
  for x being object holds x in dom f iff x = f".y1 or x = f".y2
  proof
    let x be object;
    thus x in dom f implies x = f".y1 or x = f".y2
    proof
      assume A5: x in dom f;
      then f.x in rng f by FUNCT_1:3;
      then f.x = y1 or f.x = y2  by A1, TARSKI:def 2;
      hence thesis by A5, FUNCT_1:34;
    end;
    thus thesis by A3,A4, FUNCT_1:34;
  end;
  hence thesis by TARSKI:def 2;
end;
