reserve i for Integer,
  a, b, r, s for Real;

theorem Th13:
  for S, T being non empty TopSpace, f being Function of S,T st f
  is one-to-one onto holds f is continuous iff f" is open
proof
  let S, T be non empty TopSpace, f be Function of S,T such that
A1: f is one-to-one;
  assume f is onto;
  then
A2: f qua Function" = f" by A1,TOPS_2:def 4;
A3: [#]T <> {};
  thus f is continuous implies f" is open
  proof
    assume
A4: f is continuous;
    let A be Subset of T;
    assume A is open;
    then f"A is open by A3,A4,TOPS_2:43;
    hence thesis by A1,A2,FUNCT_1:85;
  end;
  assume
A5: f" is open;
  now
    let A be Subset of T;
    assume A is open;
    then f".:A is open by A5;
    hence f"A is open by A1,A2,FUNCT_1:85;
  end;
  hence thesis by A3,TOPS_2:43;
end;
