
theorem Th2:
  for X being non empty TopSpace,
      f being Function of the carrier of X,COMPLEX holds
  ( f is continuous iff for Y being Subset of COMPLEX st Y is open holds
    f " Y is open )
proof
  let X be non empty TopSpace,
      f be Function of the carrier of X,COMPLEX;
  hereby
    assume
A1: f is continuous;
    let Y be Subset of COMPLEX;
    assume Y is open;
    then Y` is closed by CFDIFF_1:def 11;
    then
A2: f"(Y`) is closed by A1;
    f"(Y`) = (f"COMPLEX) \ f"(Y) by FUNCT_1:69
      .= ([#]X) \ f"Y by FUNCT_2:40;
    then ([#]X) \ (([#]X) \ f"(Y)) is open by A2,PRE_TOPC:def 3;
    hence f"Y is open by PRE_TOPC:3;
  end;
  assume
A3: for Y being Subset of COMPLEX st Y is open holds f"Y is open;
  let Y be Subset of COMPLEX;
  assume
A4: Y is closed;
  Y = Y``;
  then Y` is open by A4,CFDIFF_1:def 11;
  then
A5: f"(Y`) is open by A3;
  f"(Y`) = (f"COMPLEX) \ f"(Y) by FUNCT_1:69
        .= ([#]X) \ f"Y by FUNCT_2:40;
  hence f"Y is closed by A5,PRE_TOPC:def 3;
end;
