
theorem Th19:
  for NrSp be RealNormSpace,
      S be Subset of NrSp,
      T be Subset of MetricSpaceNorm NrSp
    st S = T
  holds S is compact iff T is sequentially_compact
  proof
    let NrSp be RealNormSpace,
        S be Subset of NrSp,
        T be Subset of MetricSpaceNorm NrSp;
    assume
A1: S = T;
    hereby
      assume
  A2: S is compact;
      now
        let S1 be sequence of MetricSpaceNorm NrSp;
        assume
        A3: rng S1 c= T;
        reconsider s1 = S1 as sequence of NrSp;
        consider s2 be sequence of NrSp such that
        A4: s2 is subsequence of s1 & s2 is convergent
            & lim s2 in S by A1,A2,A3;
        consider N be increasing sequence of NAT such that
        A5: s2 = s1 * N by A4,VALUED_0:def 17;
        reconsider S2 = s2 as sequence of MetricSpaceNorm NrSp;
        take S2;
        thus ex N be increasing sequence of NAT st S2 = S1 * N by A5;
        thus S2 is convergent by A4,NORMSP_2:5;
        hence lim S2 in T by A1,A4,NORMSP_2:6;
      end;
      hence T is sequentially_compact;
    end;
    assume
    A6: T is sequentially_compact;
    now
      let s1 be sequence of NrSp;
      assume
  A7: rng s1 c= S;
      reconsider S1 = s1 as sequence of MetricSpaceNorm NrSp;
      consider S2 be sequence of MetricSpaceNorm NrSp such that
  A8: (ex N be increasing sequence of NAT st S2 = S1 * N)
         & S2 is convergent & lim S2 in T by A1,A6,A7;
      consider N be increasing sequence of NAT such that
  A9: S2 = S1 * N by A8;
      reconsider s2 = S2 as sequence of NrSp;
      take s2;
      thus s2 is subsequence of s1 by A9;
      thus s2 is convergent by A8,NORMSP_2:5;
      thus lim s2 in S by A1,A8,NORMSP_2:6;
    end;
    hence S is compact;
  end;
