import Mathlib.Data.Fin.Tuple.Basic
import Mathlib.Analysis.Complex.Basic
import Mathlib.Analysis.Complex.TaylorSeries
import Mathlib.Analysis.Calculus.LineDeriv.Basic
import Mathlib.Analysis.Calculus.ContDiff.Defs
import Mathlib.Analysis.Calculus.FDeriv.Basic
import Mathlib.Analysis.Calculus.FDeriv.Symmetric


lemma l₁ (f : ℝ × ℝ → ℝ) (h : ContDiff ℝ 2 f) :
  ∀ z a b : ℝ × ℝ, (fderiv ℝ (fun w => fderiv ℝ f w) z) a b = iteratedFDeriv ℝ 2 f z ![a, b] := by
  sorry

lemma l₂ (f : ℝ × ℝ → ℝ) (h : ContDiff ℝ 2 f) :
  ∀ z a b : ℝ × ℝ, (fderiv ℝ (fun w => fderiv ℝ f w) z) a b = (fderiv ℝ (fun w ↦ (fderiv ℝ f w) a) z) b := by
  sorry