Calculating the proximal subdifferential via the quasidifferential

The relation between the proximal subdifferential and the quasidifferential for a function, which is B-differentiable (uniformly directionally differentiable) and quasidifferentiable (in the sense of Demyanov and Rubinov), is developed. On the basis of this relation, the proximal subdifferential is computed via the quasidifferential. This computation can be used in the contexts of stability for a control system with a nonsmooth Lyapunov function and of nonsmooth optimization.