Corollary 1

— Let A be a continuous derivation of the A-algebra E. For every formal power series $u \in \mathbf{A}[[\mathbf{I}]]$ the family $((D_iu)(x) \cdot \Delta x_i)_{i \in I}$ is summable and we have

This follows from Prop. 7 because the mapping $u \mapsto \Delta(u(x))$ is a continuous derivation of $\mathbf{A}[[\mathbf{I}]]$ into the $\mathbf{A}[[\mathbf{I}]]$-module E.