
theorem Th37:
  for V being RealUnitarySpace, v being Point of V, r being Real
  holds Ball(v,r) in Family_open_set(V)
proof
  let V be RealUnitarySpace;
  let v be Point of V;
  let r be Real;
  for u being Point of V st u in Ball(v,r)
   ex p being Real st p>0 &
  Ball(u,p) c= Ball(v,r) by Th35;
  hence thesis by Def5;
end;
