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
  (for r st 0 < r holds Ball(x,r) contains_almost_all_sequence S) iff
for V st x in V & V in Family_open_set X holds V contains_almost_all_sequence S
  by Th15,Th16,Th17;
