Proposition 7

Suppose that $A$ is a unital associative graded algebra and $E$ a graded $(A, A)$-bimodule. Let $d : A \to E$ be an $\varepsilon$-derivation of degree $\delta$ and $a$ a homogeneous element of $Z_\varepsilon$ of degree $\alpha$. Then the mapping $x \mapsto a(dx)$ is an $\varepsilon$-derivation of degree $\delta + \alpha$.

We write $d'(x) = a(dx)$; for $x$ homogeneous of degree $\xi$ in $A$ and $y \in A$, by virtue of (23) and (1) (no. 1),

Proposition 7 says that the K-module of $\varepsilon$-derivations of $A$ into $E$ is a graded $Z_\varepsilon$-module of type $\Delta$.