
theorem
  for NrSp be RealNormSpace,
      S be Subset of NrSp,
      T be Subset of TopSpaceNorm NrSp
    st S = T
  holds S is compact iff T is compact
  proof
    let NrSp be RealNormSpace,
        S be Subset of NrSp,
        T be Subset of TopSpaceNorm NrSp;
    assume
A1: S = T;
    reconsider W = S as Subset of MetricSpaceNorm NrSp;
    W is sequentially_compact iff T is compact by A1,Th16;
    hence thesis by Th19;
  end;
