Differential operator/Algebraic/Inductive and product condition/Fact/Proof/Exercise

Let ${}R$ denote a commutative ${}K$-algebra and let ${}E\colon R\rightarrow R$ denote a ${}K$-linear map. Show that the following statements are equivalent.

1. ${}E$ is a differential operator of order ${}\leq n$.
2. For arbitrary elements ${}f_{0},f_{1},\ldots ,f_{n}\in R$ we have

   ${}{\begin{aligned}E(f_{0}\cdots f_{n})&=\sum _{s=1}^{n}(-1)^{s+1}\sum _{I\subseteq \{0,\ldots ,n\},\,{\#\left(I\right)}=s}\prod _{i\in I}f_{i}\cdot E{\left(\prod _{i\notin I}f_{i}\right)}\\&=\sum _{I\subseteq \{0,\ldots ,n\},\,I\neq \emptyset }(-1)^{{\#\left(I\right)}+1}\prod _{i\in I}f_{i}\cdot E{\left(\prod _{i\notin I}f_{i}\right)}.\end{aligned}}$

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