Proposition 3

*Let $X_1, X_2, \ldots, X_p, Y$ be manifolds of class $C^r$, $i$ an integer of $\{1, p\}$, $m_i$ (resp. $n$) a law of composition of class $C^r$ on $X_i$ (resp. $Y$) and $u$ a mapping of class $C^r$ of $X_1 \times X_2 \times \cdots \times X_p$ into $Y$. If $u$ is distributive relative to the variable of index $i$, then $T(u)$ is distributive relative to the variable of index $i$.

The proof is analogous to that of Proposition 1.