reserve X for RealNormSpace;

theorem Th5:
  for X be RealNormSpace, S be sequence of X, St be sequence of
  MetricSpaceNorm 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
  MetricSpaceNorm X;
  assume
A1: S=St;
A2: now
    assume S is convergent;
    then consider x be Point of X such that
A3: for r be Real st 0 < r ex m be Nat st
     for n be Nat st m <= n holds ||. S.n - x.|| < r by NORMSP_1:def 6;
    reconsider xt=x as Point of MetricSpaceNorm X;
    St is_convergent_in_metrspace_to xt by A1,A3,Th4;
    hence St is convergent by METRIC_6:9;
  end;
  now
    assume St is convergent;
    then consider xt be Point of MetricSpaceNorm X such that
A4: St is_convergent_in_metrspace_to xt by METRIC_6:10;
    reconsider x=xt as Point of X;
    for r be Real st 0 < r ex m be Nat st
      for n be Nat st m <= n holds ||. S.n - x.|| < r by A1,A4,Th4;
    hence S is convergent by NORMSP_1:def 6;
  end;
  hence thesis by A2;
end;
