A Lemma for a Strong Comparison Principle of Nonlinear Parabolic Equations

In this note, we prove a lemma for a strong comparison principle of nonlinear parabolic equations. We shall prove a function which is a viscosity subsolution minus a viscosity supersolution of the equation becomes a viscosity subsolution of a parabolic equation which may not coincide with the original equation. Thanks to a strong maximum principle of nonlinear parabolic equations we have a strong comparison principle.