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 Th51:
  for M being Reflexive symmetric triangle non empty MetrStruct,
  z being Point of M holds r < 0 implies Sphere(z,r) = {}
proof
  let M be Reflexive symmetric triangle non empty MetrStruct, z be Point of
  M;
  assume
A1: r < 0;
  thus Sphere(z,r) c= {}
  proof
    let a be object;
    assume
A2: a in Sphere(z,r);
    then reconsider b = a as Point of M;
    dist(b,z) = r by A2,METRIC_1:13;
    hence thesis by A1,METRIC_1:5;
  end;
  thus thesis;
end;
