theorem
  S is_convergent_in_metrspace_to x iff for V st x in V & V in
  Family_open_set X holds V contains_almost_all_sequence S
proof
  thus S is_convergent_in_metrspace_to x implies for V st x in V & V in
  Family_open_set X holds V contains_almost_all_sequence S
  proof
    assume S is_convergent_in_metrspace_to x;
    then for r st 0 < r holds Ball(x,r) contains_almost_all_sequence S by Th15;
    hence thesis by Th16;
  end;
  thus thesis by Th17;
end;
