
theorem
  for V being RealUnitarySpace, M being Subset of TopUnitSpace V, v
  being VECTOR of V, r being Real st M = Sphere(v,r) holds M is closed
proof
  let V be RealUnitarySpace;
  let M be Subset of TopUnitSpace V;
  let v be VECTOR of V;
  let r be Real;
  reconsider B = cl_Ball(v,r), C = Ball(v,r) as Subset of TopUnitSpace V;
  assume
A1: M = Sphere(v,r);
A2: M` = B` \/ C
  proof
    hereby
      let a be object;
      assume
A3:   a in M`;
      then reconsider e = a as Point of V;
      not a in M by A3,XBOOLE_0:def 5;
      then dist(e,v) <> r by A1,BHSP_2:52;
      then dist(e,v) < r or dist(e,v) > r by XXREAL_0:1;
      then e in Ball(v,r) or not e in cl_Ball(v,r) by BHSP_2:41,48;
      then e in Ball(v,r) or e in cl_Ball(v,r)` by SUBSET_1:29;
      hence a in B` \/ C by XBOOLE_0:def 3;
    end;
    let a be object;
    assume
A4: a in B` \/ C;
    then reconsider e = a as Point of V;
    a in B` or a in C by A4,XBOOLE_0:def 3;
    then not e in cl_Ball(v,r) or e in C by XBOOLE_0:def 5;
    then dist(e,v) > r or dist(e,v) < r by BHSP_2:41,48;
    then not a in M by A1,BHSP_2:52;
    hence thesis by A4,SUBSET_1:29;
  end;
  B is closed & C is open by Th50,Th53;
  hence thesis by A2;
end;
