reserve X for RealNormSpace;

theorem Th16:
  for X be RealNormSpace, V be Subset of X, Vt be Subset of
  TopSpaceNorm X st V = Vt holds V is open iff Vt is open
proof
  let X be RealNormSpace, V be Subset of X, Vt be Subset of TopSpaceNorm X;
A1: V is open iff V` is closed by NFCONT_1:def 4;
  assume V = Vt;
  then V is open iff Vt` is closed by A1,Th15;
  hence thesis by TOPS_1:4;
end;
