Proposition 3

Let $u \in A[X]$ and let $X.Z$ be the family $(X_iZ)_{i \in I}$, of elements of the polynomial ring $A[X][Z]$. The coefficient of $Z^k$ in $u(X.Z)$ is the homogeneous component of degree $k$ of $u$, for every positive integer $k$.

It suffices to prove this Prop. in the case where $u$ is a monomial, and in this case the result is clear.