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
  cl_Ball(z,0) = {z}
proof
  thus cl_Ball(z,0) c= {z}
  proof
    let a be object;
    assume
A1: a in cl_Ball(z,0);
    then reconsider b = a as Point of M;
    dist(b,z) <= 0 by A1,METRIC_1:12;
    then dist(b,z) = 0 by METRIC_1:5;
    then b = z by METRIC_1:2;
    hence thesis by TARSKI:def 1;
  end;
  let a be object;
  assume a in {z};
  then
A2: a = z by TARSKI:def 1;
  dist(z,z) = 0 by METRIC_1:1;
  hence thesis by A2,METRIC_1:12;
end;
