 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 open 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
  & Ball(z,r) = {y where y is Point of X: ||.t-y.|| < r})
  & Ball(z,r)
    in Family_open_set(MetricSpaceNorm RUSp2RNSp X) by Th2,PCOMPS_1:29;
  hence thesis by PRE_TOPC:def 2;
end;
