reserve r for Real;

theorem
  for M being MetrStruct, p being Element of M holds
  Sphere(p,r) \/ Ball(p,r) = cl_Ball(p,r)
proof
  let M be MetrStruct,p be Element of M;
  Sphere(p,r) c= cl_Ball(p,r) & Ball(p,r) c= cl_Ball(p,r) by Th14,Th15;
  hence Sphere(p,r) \/ Ball(p,r) c= cl_Ball(p,r) by XBOOLE_1:8;
  per cases;
  suppose
A1:  M is non empty;
    now
      let x be Element of M;
      assume x in cl_Ball(p,r);
      then
A2:   dist(p,x)<=r by Th12;
      now
        per cases by A2,XXREAL_0:1;
        case
          dist(p,x)<r;
          hence x in Ball(p,r) by A1,Th11;
        end;
        case
          dist(p,x)=r;
          hence x in Sphere(p,r) by A1,Th13;
        end;
      end;
      hence x in Sphere(p,r) \/ Ball(p,r) by XBOOLE_0:def 3;
    end;
    hence thesis by SUBSET_1:2;
  end;
  suppose
A3: M is empty;
    then Ball(p,r) is empty & cl_Ball(p,r) is empty;
    hence thesis by A3,Def16;
  end;
end;
