theorem extracted_formal_statement_194.{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) (ω : (Ω × Ω) × (Fin n → Ω × Ω)) (σ : Signs (n + 1))
  (h_original_goal :
    ⨆ I, ∑ i, ↑↑(σ ↑↑i) * (f I (X (ω.2 i).1) - f I (X (ω.2 i).2)) + (↑↑(σ ↑n) * (f I (X ω.1.1) - f I (X ω.1.2)) + c I) =
      ⨆ I, ∑ i, ↑↑(σ i) * (f I (X (Fin.snoc ω.2 ω.1 i).1) - f I (X (Fin.snoc ω.2 ω.1 i).2)) + c I)
  (i : ι) :
  ∑ i_1, ↑↑(σ ↑↑i_1) * (f i (X (ω.2 i_1).1) - f i (X (ω.2 i_1).2)) +
      (↑↑(σ ↑n) * (f i (X ω.1.1) - f i (X ω.1.2)) + c i) =
    ∑ i_1, ↑↑(σ i_1) * (f i (X (Fin.snoc ω.2 ω.1 i_1).1) - f i (X (Fin.snoc ω.2 ω.1 i_1).2)) + c i := sorry

State before:
case h.h.e_a.e_f.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
ω : (Ω × Ω) × (Fin n → Ω × Ω)
σ : Signs (n + 1)
⊢ ⨆ I, ∑ i, ↑↑(σ ↑↑i) * (f I (X (ω.2 i).1) - f I (X (ω.2 i).2)) + (↑↑(σ ↑n) * (f I (X ω.1.1) - f I (X ω.1.2)) + c I) =
    ⨆ I, ∑ i, ↑↑(σ i) * (f I (X (Fin.snoc ω.2 ω.1 i).1) - f I (X (Fin.snoc ω.2 ω.1 i).2)) + c I

Tactic: apply iSup_congr

State after:
[case h.h.e_a.e_f.h.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
 ω : (Ω × Ω) × (Fin n → Ω × Ω)
 σ : Signs (n + 1)
 ⊢ ∀ (i : ι),
     ∑ i_1, ↑↑(σ ↑↑i_1) * (f i (X (ω.2 i_1).1) - f i (X (ω.2 i_1).2)) +
         (↑↑(σ ↑n) * (f i (X ω.1.1) - f i (X ω.1.2)) + c i) =
       ∑ i_1, ↑↑(σ i_1) * (f i (X (Fin.snoc ω.2 ω.1 i_1).1) - f i (X (Fin.snoc ω.2 ω.1 i_1).2)) + c i]

Relevant lemmas:
iSup_congr : ∀ {α : Type u_1} {ι : Sort u_4} [inst : SupSet α] {f g : ι → α},
  (∀ (i : ι), f i = g i) → ⨆ i, f i = ⨆ i, g i