 reserve X for RealUnitarySpace;
 reserve x, y, y1, y2 for Point of X;

theorem
  for X be RealUnitarySpace, x be Point of X, r be Real
  holds {y where y is
  Point of X: ||.x-y.|| <= r} is closed Subset of TopSpaceNorm RUSp2RNSp X
proof
  let X be RealUnitarySpace, x be Point of X, r be Real;
  reconsider z=x as Element of MetricSpaceNorm RUSp2RNSp X;
  ex t be Point of X st t = x & cl_Ball(z,r) = {y where y is Point of X:
  ||.t-y.|| <= r} by Th3;
  hence thesis by TOPREAL6:57;
end;
