reserve a, b for Real,
  r for Real,
  rr for Real,
  i, j, n for Nat,
  M for non empty MetrSpace,
  p, q, s for Point of TOP-REAL 2,
  e for Point of Euclid 2,
  w for Point of Euclid n,
  z for Point of M,
  A, B for Subset of TOP-REAL n,
  P for Subset of TOP-REAL 2,
  D for non empty Subset of TOP-REAL 2;
reserve a, b for Real;
reserve a, b for Real;
reserve r for Real;

theorem
  for M being Reflexive symmetric triangle non empty MetrStruct, z
  being Point of M holds r < 0 implies cl_Ball(z,r) = {}
proof
  let M be Reflexive symmetric triangle non empty MetrStruct, z be Point of
  M;
A1: Sphere(z,r) \/ Ball(z,r) = cl_Ball(z,r) by METRIC_1:16;
  assume
A2: r < 0;
  then Ball(z,r) = {} by TBSP_1:12;
  hence thesis by A2,A1,Th51;
end;
