English
If g ∘ f is C^n within s at x and the hypotheses hold, then the i-th iterated derivative equals the Taylor composition up to i, provided eventual membership holds.
Русский
Если композиция гладкая в i-й производной внутри s, то i-я итеративная производная совпадает с композицией Тейлора до i, при условии eventual membership.
LaTeX
$$$\forall t,\; \text{(conditions stated)} \Rightarrow \ iteratedFDerivWithin 𝕜 i (g \circ f) s x = (ftaylorSeriesWithin 𝕜 g t (f x)).taylorComp (ftaylorSeriesWithin 𝕜 f s x) i$$$
Lean4
/-- The first projection on a domain in a product is `C^∞`. -/
@[fun_prop]
theorem contDiffOn_fst {s : Set (E × F)} : ContDiffOn 𝕜 n (Prod.fst : E × F → E) s :=
ContDiff.contDiffOn contDiff_fst
