
theorem Th4:
  for T,S being non empty TopSpace, f being Function of T,S, B
  being Subset-Family of S st f is continuous & B is open holds f"B is open
proof
  let T,S be non empty TopSpace;
  let f be Function of T,S;
  let B be Subset-Family of S;
  assume that
A1: f is continuous and
A2: B is open;
  for P being Subset of T holds P in f"B implies P is open
  proof
    let P be Subset of T;
    assume
A3: P in f"(B);
    thus P is open
    proof
      consider C being Subset of S such that
A4:   C in B and
A5:   P = f"C by A3,FUNCT_2:def 9;
      reconsider C as Subset of S;
      [#]S <> {} & C is open by A2,A4,TOPS_2:def 1;
      hence thesis by A1,A5,TOPS_2:43;
    end;
  end;
  hence thesis by TOPS_2:def 1;
end;
