English
If a LocallyFinite family has ContMDiffWithinAt at x, then their finprod is ContMDiffWithinAt at x.
Русский
Если локально конечная совокупность имеет ContMDiffWithinAt в x, то их финпроизведение имеет ContMDiffWithinAt в x.
LaTeX
$$$ LocallyFinite (fun i => mulSupport (f_i)) \\\\land \\\\forall i, ContMDiffWithinAt I' I n (f_i) s x \\\\Rightarrow ContMDiffWithinAt I' I n (finprod i, f_i) s x$$$
Lean4
/-- The composition of `C^n` functions within domains at points is `C^n`. -/
theorem comp' {t : Set M'} {g : M' → M''} (x : M) (hg : ContMDiffWithinAt I' I'' n g t (f x))
(hf : ContMDiffWithinAt I I' n f s x) : ContMDiffWithinAt I I'' n (g ∘ f) (s ∩ f ⁻¹' t) x :=
hg.comp x (hf.mono inter_subset_left) inter_subset_right
