theorem extracted_formal_statement_150.{w, v, u} {Z : Type w} {ι : Type v} {f : ι → Z → ℝ} {Ω : Type u}
  [inst : MeasurableSpace Ω] {X : Ω → Z} {μ : Measure Ω} [inst_1 : IsProbabilityMeasure μ] {n : ℕ} (ω : Ω × Ω)
  [inst_2 : Countable ι] [inst_3 : Nonempty ι] (h𝓕 : ∀ (I : ι), Measurable (f I ∘ X)) {b : ℝ}
  (h𝓕' : ∀ (I : ι) (z : Z), |f I z| ≤ b) {c : ι → ℝ} {C : ℝ} (hC : ∀ (I : ι), |c I| ≤ C) :
  let μ2n := Measure.pi fun x => μ.prod μ;
  2⁻¹ *
        ((∫ (x : Fin n → Ω × Ω),
            (fun ω' =>
                2⁻¹ ^ n *
                  ∑ σ, ⨆ I, ∑ i, ↑↑(σ i) * (f I (X (ω' i).1) - f I (X (ω' i).2)) + (f I (X ω.2) - f I (X ω.1) + c I))
              x ∂Measure.pi fun x => μ.prod μ) +
          ∫ (x : Fin n → Ω × Ω),
            (fun ω' =>
                2⁻¹ ^ n *
                  ∑ σ, ⨆ I, ∑ i, ↑↑(σ i) * (f I (X (ω' i).1) - f I (X (ω' i).2)) + (f I (X ω.1) - f I (X ω.2) + c I))
              x ∂Measure.pi fun x => μ.prod μ) =
      2⁻¹ *
        ∫ (x : Fin n → Ω × Ω),
          (fun ω' =>
              (2⁻¹ ^ n *
                  ∑ σ, ⨆ I, ∑ i, ↑↑(σ i) * (f I (X (ω' i).1) - f I (X (ω' i).2)) + (f I (X ω.2) - f I (X ω.1) + c I)) +
                2⁻¹ ^ n *
                  ∑ σ, ⨆ I, ∑ i, ↑↑(σ i) * (f I (X (ω' i).1) - f I (X (ω' i).2)) + (f I (X ω.1) - f I (X ω.2) + c I))
            x ∂μ2n →
    ((∫ (x : Fin n → Ω × Ω),
          (fun ω' =>
              2⁻¹ ^ n *
                ∑ σ, ⨆ I, ∑ i, ↑↑(σ i) * (f I (X (ω' i).1) - f I (X (ω' i).2)) + (f I (X ω.2) - f I (X ω.1) + c I))
            x ∂Measure.pi fun x => μ.prod μ) +
        ∫ (x : Fin n → Ω × Ω),
          (fun ω' =>
              2⁻¹ ^ n *
                ∑ σ, ⨆ I, ∑ i, ↑↑(σ i) * (f I (X (ω' i).1) - f I (X (ω' i).2)) + (f I (X ω.1) - f I (X ω.2) + c I))
            x ∂Measure.pi fun x => μ.prod μ) =
      ∫ (x : Fin n → Ω × Ω),
        (fun ω' =>
            (2⁻¹ ^ n *
                ∑ σ, ⨆ I, ∑ i, ↑↑(σ i) * (f I (X (ω' i).1) - f I (X (ω' i).2)) + (f I (X ω.2) - f I (X ω.1) + c I)) +
              2⁻¹ ^ n *
                ∑ σ, ⨆ I, ∑ i, ↑↑(σ i) * (f I (X (ω' i).1) - f I (X (ω' i).2)) + (f I (X ω.1) - f I (X ω.2) + c I))
          x ∂μ2n := sorry

State before:
Z : Type w
ι : Type v
f : ι → Z → ℝ
Ω : Type u
inst✝³ : MeasurableSpace Ω
X : Ω → Z
μ : Measure Ω
inst✝² : IsProbabilityMeasure μ
n : ℕ
ω : Ω × Ω
inst✝¹ : Countable ι
inst✝ : Nonempty ι
h𝓕 : ∀ (I : ι), Measurable (f I ∘ X)
b : ℝ
h𝓕' : ∀ (I : ι) (z : Z), |f I z| ≤ b
c : ι → ℝ
C : ℝ
hC : ∀ (I : ι), |c I| ≤ C
μ2n : Measure (Fin n → Ω × Ω) := Measure.pi fun x => μ.prod μ
⊢ 2⁻¹ *
      ((∫ (x : Fin n → Ω × Ω),
          (fun ω' =>
              2⁻¹ ^ n *
                ∑ σ, ⨆ I, ∑ i, ↑↑(σ i) * (f I (X (ω' i).1) - f I (X (ω' i).2)) + (f I (X ω.2) - f I (X ω.1) + c I))
            x ∂Measure.pi fun x => μ.prod μ) +
        ∫ (x : Fin n → Ω × Ω),
          (fun ω' =>
              2⁻¹ ^ n *
                ∑ σ, ⨆ I, ∑ i, ↑↑(σ i) * (f I (X (ω' i).1) - f I (X (ω' i).2)) + (f I (X ω.1) - f I (X ω.2) + c I))
            x ∂Measure.pi fun x => μ.prod μ) =
    2⁻¹ *
      ∫ (x : Fin n → Ω × Ω),
        (fun ω' =>
            (2⁻¹ ^ n *
                ∑ σ, ⨆ I, ∑ i, ↑↑(σ i) * (f I (X (ω' i).1) - f I (X (ω' i).2)) + (f I (X ω.2) - f I (X ω.1) + c I)) +
              2⁻¹ ^ n *
                ∑ σ, ⨆ I, ∑ i, ↑↑(σ i) * (f I (X (ω' i).1) - f I (X (ω' i).2)) + (f I (X ω.1) - f I (X ω.2) + c I))
          x ∂μ2n

Tactic: apply congr_arg

State after:
[case h
 Z : Type w
 ι : Type v
 f : ι → Z → ℝ
 Ω : Type u
 inst✝³ : MeasurableSpace Ω
 X : Ω → Z
 μ : Measure Ω
 inst✝² : IsProbabilityMeasure μ
 n : ℕ
 ω : Ω × Ω
 inst✝¹ : Countable ι
 inst✝ : Nonempty ι
 h𝓕 : ∀ (I : ι), Measurable (f I ∘ X)
 b : ℝ
 h𝓕' : ∀ (I : ι) (z : Z), |f I z| ≤ b
 c : ι → ℝ
 C : ℝ
 hC : ∀ (I : ι), |c I| ≤ C
 μ2n : Measure (Fin n → Ω × Ω) := Measure.pi fun x => μ.prod μ
 ⊢ ((∫ (x : Fin n → Ω × Ω),
         (fun ω' =>
             2⁻¹ ^ n *
               ∑ σ, ⨆ I, ∑ i, ↑↑(σ i) * (f I (X (ω' i).1) - f I (X (ω' i).2)) + (f I (X ω.2) - f I (X ω.1) + c I))
           x ∂Measure.pi fun x => μ.prod μ) +
       ∫ (x : Fin n → Ω × Ω),
         (fun ω' =>
             2⁻¹ ^ n *
               ∑ σ, ⨆ I, ∑ i, ↑↑(σ i) * (f I (X (ω' i).1) - f I (X (ω' i).2)) + (f I (X ω.1) - f I (X ω.2) + c I))
           x ∂Measure.pi fun x => μ.prod μ) =
     ∫ (x : Fin n → Ω × Ω),
       (fun ω' =>
           (2⁻¹ ^ n *
               ∑ σ, ⨆ I, ∑ i, ↑↑(σ i) * (f I (X (ω' i).1) - f I (X (ω' i).2)) + (f I (X ω.2) - f I (X ω.1) + c I)) +
             2⁻¹ ^ n *
               ∑ σ, ⨆ I, ∑ i, ↑↑(σ i) * (f I (X (ω' i).1) - f I (X (ω' i).2)) + (f I (X ω.1) - f I (X ω.2) + c I))
         x ∂μ2n]

Relevant lemmas:
congr_arg : ∀ {α : Sort u} {β : Sort v} {a₁ a₂ : α} (f : α → β), a₁ = a₂ → f a₁ = f a₂