theorem extracted_formal_statement_68.{u_1, w, u} {Ω : Type u} [inst : MeasurableSpace Ω] {Z : Type w} {n : ℕ}
  {ι : Type u_1} {f : ι → Z → ℝ} {μ : Measure Ω} [inst_1 : IsProbabilityMeasure μ] {X : Ω → Z} [inst_2 : Nonempty ι]
  [inst_3 : Countable ι] (hf : ∀ (i : ι), Measurable (f i ∘ X)) (x : Fin n → Ω) (i : ι) (k : Fin n) :
  f i (X (x k)) - ∫ (x : Ω), f i (X x) ∂μ =
    f i (X (x k)) - ∫ (x : Fin n → Ω), f i (X (x k)) ∂Measure.pi fun x => μ := sorry

State before:
case e_f.h.e_s.h.e_a.e_f.h
Ω : Type u
inst✝³ : MeasurableSpace Ω
Z : Type w
n : ℕ
ι : Type u_1
f : ι → Z → ℝ
μ : Measure Ω
inst✝² : IsProbabilityMeasure μ
X : Ω → Z
inst✝¹ : Nonempty ι
inst✝ : Countable ι
hf : ∀ (i : ι), Measurable (f i ∘ X)
x✝ : Fin n → Ω
i : ι
k : Fin n
⊢ f i (X (x✝ k)) - ∫ (x : Ω), f i (X x) ∂μ = f i (X (x✝ k)) - ∫ (x : Fin n → Ω), f i (X (x k)) ∂Measure.pi fun x => μ

Tactic: apply congrArg
    have : (Measure.map (fun f ↦ f k) (μⁿ))[fun y ↦ f i (X y)] = μⁿ[fun x : Fin n → Ω ↦ f i (X (x k))] :=
  integral_map (measurable_pi_apply k).aemeasurable (Measurable.aestronglyMeasurable (hf i))
    rw [← this, pi_map_eval k]

State after:

Relevant lemmas:
error