I'm reading Representation of Algebraic Groups of Jantzen about Distribution of Algebra. In chapter 7, page 103, 7.11, the author is stating that if $G$ is a group scheme over $k$, then any $G$-module $M$ induces a natural $Dist(G)$-module structure: One set for each $\mu \in Dist(G), m \in M$: $$\mu m = (id_M \otimes \mu)\Delta_M(m)$$. To state that this really induces a $Dist(G)$-module structure, we need to confirm that $\mu(\nu m) = (\mu \nu)m$ $\forall m \in M, \mu, \nu \in Dist(G)$. Here is my trying computation:
$$\mu(\nu m) = (id_M \otimes \mu)(\Delta_M(\nu m)) = (id_M \otimes \mu) \Delta_M(id_M \otimes \nu) \Delta_M(m)$$ $$(\mu \nu)m=(id_M \otimes \mu \nu)\Delta_M(m) = (id_M \otimes ((\mu \otimes \nu) \Delta_G))\Delta_M(m)$$
This is exactly where I got stuck. I don't know how to prove that these two equations are the same. The author suggests to use 2.8(2) and (3), but to use them, in place of $\mu, \nu$, we must have $\Delta_G$. Any suggestion or hint is really appreciated. Thanks