Proposition 1

Let $E_1, E_2, \ldots, E_n$ and F be commutative monoids written additively and let u be a mapping of $E_1 \times \cdots \times E_n$ into F, which is distributive with respect to all the variables. For $i = 1, 2, \ldots, n$, let $L_i$ be a non-empty finite set and $(x_{i,\lambda})_{\lambda \in L_i}$ a family of elements of $E_i$. Let $y_i = \sum_{\lambda \in L_i} x_{i,\lambda}$ for $i = 1, 2, \ldots, n$. Then

(2)

the sum being taken over all sequences $\alpha = (\alpha_1, \ldots, \alpha_n)$ belonging to $L_1 \times \cdots \times L_n$.

We argue by induction on n, the case $n = 1$ following from formula (2) of § 1, no. 2. From the same reference

(3)

for $y_n = \sum_{\alpha_n \in L_n} x_{n,\alpha_n}$ and the mapping $z \mapsto u(y_1, \ldots, y_{n-1}, z)$ of $E_n$ into $F$ is a magma homomorphism. By the induction hypothesis applied to the distributive mappings $(z_1, \ldots, z_{n-1}) \mapsto u(z_1, \ldots, z_{n-1}, x_n, \alpha_n)$ from $E_1 \times \cdots \times E_{n-1}$ to $F$,

(4)

the sum being taken over the sequences $(\alpha_1, \ldots, \alpha_{n-1})$ belonging to $M = L_1 \times \cdots \times L_{n-1}$. Now $L_1 \times \cdots \times L_n = M \times L_n$; writing we have (5) by formula (7) of § 1, no. 5. (2) follows immediately from (3), (4) and (5).