Faà 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. 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 both forms.