Superposition principle on the viscosity solutions of infinity Laplace equations

We consider the sum of the solutions of two infinity Laplace equations in disjoint variables. We prove that the superposed function is a viscosity solution of the infinity Laplace equation in the extension domains with the sum of inhomogeneous terms if one of the solutions is in the sense of viscosity and the other is in the classical sense. We also construct a counterexample to show that the conclusion may not be true if both of the solutions are merely in the viscosity sense.