Corollary 1

Let U be an open set of $\mathbf{R}^n$. Let $k\in \mathbf{N}$ and $f$ be a mapping of $X\times U$ into E satisfying the following conditions:

(i) For every $t\in U$, the mapping $x\mapsto f(x, t)$ of X into E is $\nu$-integrable;

(ii) For every $x\in X$, the mapping $t\mapsto f(x, t)$ of U into E is of class $C^k$, with partial derivatives denoted by $\partial^{\alpha}f(x, t)$ for every $\alpha \in \mathbf{N}^n$ such that $|\alpha |\leqslant k$;

(iii) There exists a $\nu$-integrable function $g$ on X such that for every $\alpha \in \mathbf{N}^n$ satisfying $|\alpha |\leqslant k$ and for every $(x, t)\in X\times U$, one has $\|\partial^{\alpha}f(x, t)\|\leqslant g(x)$.

Then the mapping F of U into E defined by

is of class $C^k$ in U, and for every $\alpha \in \mathbf{N}^n$ satisfying $|\alpha |\leqslant k$ and every $t\in U$, one has

This follows from the proposition by induction on $k$.