reserve X for RealUnitarySpace;
reserve x, y, z, g, g1, g2 for Point of X;
reserve a, q, r for Real;
reserve seq, seq1, seq2, seq9 for sequence of X;
reserve k, n, m, m1, m2 for Nat;

theorem
  Ball(x,r) \/ Sphere(x,r) = cl_Ball(x,r)
proof
  now
    let y;
    assume y in cl_Ball(x,r);
    then
A1: ||.x - y.|| <= r by Th47;
    now
      per cases by A1,XXREAL_0:1;
      case
        ||.x - y.|| < r;
        hence y in Ball(x,r);
      end;
      case
        ||.x - y.|| = r;
        hence y in Sphere(x,r);
      end;
    end;
    hence y in Ball(x,r) \/ Sphere(x,r) by XBOOLE_0:def 3;
  end;
  then
A2: cl_Ball(x,r) c= Ball(x,r) \/ Sphere(x,r) by SUBSET_1:2;
  Ball(x,r) c= cl_Ball(x,r) & Sphere(x,r) c= cl_Ball(x,r) by Th54,Th55;
  then Ball(x,r) \/ Sphere(x,r) c= cl_Ball(x,r) by XBOOLE_1:8;
  hence thesis by A2,XBOOLE_0:def 10;
end;
