reserve X for RealNormSpace;

theorem Th13:
  for X be RealNormSpace, S be sequence of X, St be sequence of
  TopSpaceNorm X st S = St holds St is convergent iff S is convergent
proof
  let X be RealNormSpace, S be sequence of X, St be sequence of TopSpaceNorm X;
  reconsider Sm=St as sequence of MetricSpaceNorm X;
A1: St is convergent iff Sm is convergent by FRECHET2:29;
  assume S = St;
  hence thesis by A1,Th5;
end;
