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 Th58:
  for A being Subset of TopSpaceMetr M st A = Sphere(z,r) holds A is closed
proof
  let A be Subset of TopSpaceMetr M such that
A1: A = Sphere(z,r);
  reconsider B = cl_Ball(z,r), C = Ball(z,r) as Subset of TopSpaceMetr M by
TOPMETR:12;
A2: (cl_Ball(z,r))` = B` by TOPMETR:12;
A3: A` = B` \/ C
  proof
    hereby
      let a be object;
      assume
A4:   a in A`;
      then reconsider e = a as Point of M by TOPMETR:12;
      not a in A by A4,XBOOLE_0:def 5;
      then dist(e,z) <> r by A1,METRIC_1:13;
      then dist(e,z) < r or dist(e,z) > r by XXREAL_0:1;
      then e in Ball(z,r) or not e in cl_Ball(z,r) by METRIC_1:11,12;
      then e in Ball(z,r) or e in cl_Ball(z,r)` by SUBSET_1:29;
      hence a in B` \/ C by A2,XBOOLE_0:def 3;
    end;
    let a be object;
    assume
A5: a in B` \/ C;
    then reconsider e = a as Point of M by TOPMETR:12;
    a in B` or a in C by A5,XBOOLE_0:def 3;
    then not e in cl_Ball(z,r) or e in C by XBOOLE_0:def 5;
    then dist(e,z) > r or dist(e,z) < r by METRIC_1:11,12;
    then not a in A by A1,METRIC_1:13;
    hence thesis by A5,SUBSET_1:29;
  end;
A6: C is open by TOPMETR:14;
  B` is open by Lm1;
  hence thesis by A3,A6,TOPS_1:3;
end;
