Fa\`a di Bruno is Taylor Composition

We prove that reduced Taylor polynomials compose: for $C^k$ maps $\phi: E \to F$ and $\psi: F \to G$ between Banach spaces, $T_{\ast}^k(\psi\circ\phi;\, x) = \pi_{\leq k}\bigl(T_{\ast}^k(\psi;\, y) \circ T_{\ast}^k(\phi;\, x)\bigr)$ where $y = \phi(x)$.
The proof is a direct estimate of the Peano remainder and requires no combinatorics or partition arguments. From this we derive the multivariate Faà di Bruno formula in partition form (Levy 2006), by polarization, and in multi-index form (Constantine-Savits 1996) by coefficient extraction.
As an application we give a higher-order product rule in three forms.