reserve X for MetrSpace,
  x,y,z for Element of X,
  A for non empty set,
  G for Function of [:A,A:],REAL,
  f for Function,
  k,n,m,m1,m2 for Nat,
  q,r for Real;
reserve X for non empty MetrSpace,
  x,y for Element of X,
  V for Subset of X,
  S,S1,T for sequence of X,
  Nseq for increasing sequence of NAT;

theorem Th16:
  (for r st 0 < r holds Ball(x,r) contains_almost_all_sequence S)
  implies for V st x in V & V in Family_open_set X holds V
  contains_almost_all_sequence S
proof
  assume
A1: for r st 0 < r holds Ball(x,r) contains_almost_all_sequence S;
A2: for r st 0 < r ex m st for n st m <= n holds S.n in Ball(x,r)
  by A1,Def5;
  thus for V st x in V & V in Family_open_set X holds V
  contains_almost_all_sequence S
  proof
    let V;
    assume x in V & V in Family_open_set X;
    then consider q such that
A3: 0 < q and
A4: Ball(x,q) c= V by PCOMPS_1:def 4;
    consider m1 such that
A5: for n st m1 <= n holds S.n in Ball(x,q) by A2,A3;
    for n st m1 <= n holds S.n in V by A4,A5;
    hence thesis;
  end;
end;
