reserve X for RealNormSpace;

theorem Th12:
  for X be RealNormSpace, S be sequence of X, St be sequence of
TopSpaceNorm X, x be Point of X, xt be Point of TopSpaceNorm X st S = St & x =
  xt holds St is_convergent_to xt iff
 for r be Real st 0 < r ex m be Nat
   st for n be Nat st m <= n holds ||. S.n - x.|| < r
proof
  let X be RealNormSpace, S be sequence of X, St be sequence of TopSpaceNorm X
  , x be Point of X, xt be Point of TopSpaceNorm X;
  assume that
A1: S=St and
A2: x=xt;
  reconsider Sm=St as sequence of MetricSpaceNorm X;
  reconsider xm=x as Point of MetricSpaceNorm X;
  St is_convergent_to xt iff Sm is_convergent_in_metrspace_to xm by A2,
FRECHET2:28;
  hence thesis by A1,Th4;
end;
