Proposition 4

Let $d : A \to E$ be an $\varepsilon$-derivation of degree $\delta$. Suppose that A has a unit element 1 and let $x$ be a homogeneous element of A with an inverse $x^{-1}$ in A. Then

We have $d(xx^{-1}) = d(1) = 0$ (Proposition 3), whence

which proves the first formula of (19). On the other hand, $x^{-1}$ is homogeneous of degree $-\deg(x)$ and $\varepsilon_{\delta, \deg(x)} = \varepsilon_{\delta, -\deg(x)}$ by formulae (1) of no. 1; writing $d(x^{-1}x) = 0$, the second formula of (19) is obtained similarly.