reserve r, s, t, g for Real,

          r3, r1, r2, q3, p3 for Real;
reserve T for TopStruct,
  f for RealMap of T;

theorem Th8:
  f is continuous iff for Y being Subset of REAL st Y is open holds f"Y is open
proof
  hereby
    assume
A1: f is continuous;
    let Y be Subset of REAL;
    assume Y is open;
    then Y` is closed;
    then
A2: f"(Y`) is closed by A1;
    f"(Y`) = (f"REAL) \ f"(Y) by FUNCT_1:69
      .= ([#]T) \ f"Y by FUNCT_2:40;
    then ([#]T) \ (([#]T) \ 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 REAL st Y is open holds f"Y is open;
  let Y be Subset of REAL;
  assume Y is closed;
  then Y` is open;
  then
A4: f"(Y`) is open by A3;
  f"(Y`) = (f"REAL) \ f"(Y) by FUNCT_1:69
    .= ([#]T) \ f"Y by FUNCT_2:40;
  hence thesis by A4,PRE_TOPC:def 3;
end;
