Corollary

— Let $f \in A[X_1, ..., X_p]$ and $g_i \in A[Y_1, ..., Y_q]$ for $1 \leq i \leq p$, and write $h = f(g_1, ..., g_p)$, then for $1 \leq j \leq q$ we have

This is the special case $E = A[Y_1, ..., Y_r]$, $x_i = g_i$ and $D = \partial / \partial Y_j$ of Prop. 4.

Let $X = (X_i)_{i \in I}$, $Y = (Y_i)_{i \in I}$ be two disjoint families of indeterminates, and write $X + Y$ for the family $(X_i + Y_i)_{i \in I}$. Given $u \in A[X]$, consider the element $u(X + Y)$ of $A[X, Y]$. For $\nu \in \mathbf{N}^{(I)}$ we denote by $A^\nu u$ the coefficient of $Y_\nu$ in $u(X + Y)$, considered as polynomial in the $Y_i$ with coefficients in $A[X]$. By definition we have $A^\nu u \in A[X]$ and

(Here and in the rest of this No., the summations are extended over the index set $\mathbf{N}^{(I)}$ unless the contrary is said.)

Let $a \in A'$, then on substituting $a$ for $X$ and $X - a$ for $Y$ in (9) we obtain

In particular we have

If $u, v \in A[X]$, we have

hence

Let $Z = (Z_i)_{i \in I}$ be another family of indeterminates.

We have:

hence by I, p. 99, Cor. 2:

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