
theorem
  for X, Y being non empty TopStruct, f being Function of X, Y, g being
  Function of Y, X st f = id X & g = id X & f is continuous & g is continuous
  holds the TopStruct of X = the TopStruct of Y
proof
  let X, Y be non empty TopStruct, f be Function of X, Y, g be Function of Y,
  X such that
A1: f = id X and
A2: g = id X and
A3: f is continuous and
A4: g is continuous;
A5: the carrier of X = dom f by FUNCT_2:def 1
    .= the carrier of Y by A2,FUNCT_2:def 1;
A6: [#]Y <> {};
A7: [#]X <> {};
  the topology of X = the topology of Y
  proof
    hereby
      let A be object;
      assume
A8:   A in the topology of X;
      then reconsider B = A as Subset of X;
      B is open by A8;
      then
A9:   g"B is open by A4,A7,TOPS_2:43;
      g"B = B by A2,FUNCT_2:94;
      hence A in the topology of Y by A9;
    end;
    let A be object;
    assume
A10: A in the topology of Y;
    then reconsider B = A as Subset of Y;
    B is open by A10;
    then
A11: f"B is open by A3,A6,TOPS_2:43;
    f"B = B by A1,A5,FUNCT_2:94;
    hence thesis by A11;
  end;
  hence thesis by A5;
end;
