theorem
  f is continuous iff for O being open Subset of NTY holds
  f"O is open Subset of NTX
  proof
    hereby
      assume f is continuous;
      then for A being Subset of NTX holds f.:(Cl A) c= Cl(f.:A)
        by Lm13;
      then for S being closed Subset of NTY holds f"S is closed Subset of NTX
        by Lm23;
      hence for O being open Subset of NTY holds f"O is open Subset of NTX
        by Lm24;
    end;
    assume for O being open Subset of NTY holds f"O is open Subset of NTX;
    hence f is continuous by Lm26;
  end;
