English
If two functions agree on a set s in a punctured neighborhood around x, then their second iterated derivatives agree at x after congruence with respect to the punctured neighborhood.
Русский
Если две функции совпадают на множестве s в просветлении вокруг x, их вторые итерационные производные совпадают в x после конгруэнции по просветлению.
LaTeX
$$$IteratedFDerivWithin_{\mathbb{K}}^{(2)} f_{1} s x = IteratedFDerivWithin_{\mathbb{K}}^{(2)} f s x$$$
Lean4
/-- On a set of unique differentiability, the second derivative is obtained by taking the
derivative of the derivative. -/
theorem iteratedFDerivWithin_two_apply (f : E → F) {z : E} (hs : UniqueDiffOn 𝕜 s) (hz : z ∈ s) (m : Fin 2 → E) :
iteratedFDerivWithin 𝕜 2 f s z m = fderivWithin 𝕜 (fderivWithin 𝕜 f s) s z (m 0) (m 1) :=
by
simp only [iteratedFDerivWithin_succ_apply_right hs hz]
rfl
