theorem extracted_formal_statement_187.{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_1 : ℕ}
  [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) (n : ℕ)
  (ih :
    ∀ (c : ι → ℝ),
      (∃ C, ∀ (I : ι), |c I| ≤ C) →
        (∫ (x : Fin n → Ω × Ω),
            (fun ω => ⨆ I, ∑ i, (f I (X (ω i).1) - f I (X (ω i).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)) + c I)
              x ∂Measure.pi fun x => μ.prod μ)
  (c : ι → ℝ) (C : ℝ) (hC : ∀ (I : ι), |c I| ≤ C) :
  let μ2n := Measure.pi fun x => μ.prod μ;
  ∫ (x : Ω × Ω),
      (fun ω =>
          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 ∂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.1) - f I (X ω.2) + c I))
                  x ∂μ2n))
        x ∂μ.prod μ =
    2⁻¹ *
      ∫ (x : Ω × Ω),
        (fun ω =>
            ∫ (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 ∂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.1) - f I (X ω.2) + c I))
                  x ∂μ2n)
          x ∂μ.prod μ := sorry

State before:
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
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
n : ℕ
ih :
  ∀ (c : ι → ℝ),
    (∃ C, ∀ (I : ι), |c I| ≤ C) →
      (∫ (x : Fin n → Ω × Ω),
          (fun ω => ⨆ I, ∑ i, (f I (X (ω i).1) - f I (X (ω i).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)) + c I)
            x ∂Measure.pi fun x => μ.prod μ
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 : Ω × Ω),
      (fun ω =>
          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 ∂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.1) - f I (X ω.2) + c I))
                  x ∂μ2n))
        x ∂μ.prod μ =
    2⁻¹ *
      ∫ (x : Ω × Ω),
        (fun ω =>
            ∫ (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 ∂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.1) - f I (X ω.2) + c I))
                  x ∂μ2n)
          x ∂μ.prod μ

Tactic: apply integral_mul_left

State after:

Relevant lemmas:
MeasureTheory.integral_mul_left : ∀ {α : Type u_1} {m : MeasurableSpace α} {μ : Measure α} {L : Type u_6}
  [inst : RCLike L] (r : L) (f : α → L), ∫ (a : α), r * f a ∂μ = r * ∫ (a : α), f a ∂μ