Let $I$ be the ideal of smooth germs vanishing at zero. Let $I^{k+1}$ be the ideal generated by $(k+1)$-fold product of such germs. Write $F_k$ for the ideal of $k$-flat germs at zero.
By the product rule $I^{k+1}\subset F_k$. The converse inclusion is usually called Hadamard's lemma, and asserts for $k=1$ that a smooth germ $f$ may be written $f(x) = f(0) + \sum_{i=1}^n x_i g_i(x)$ with smooth $g_i$.
The only proof I know uses integration.
Question. Is it possible to prove Hadamard's lemma (the equality $I^{k+1}= F_k$) without integration?
