theorem extracted_formal_statement_307.{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)) {b : ℝ} (hf' : ∀ (i : ι) (z : Z), |f i z| ≤ b)
  (ω : Fin n → Ω)
  (this :
    (∫ (x : Fin n → Ω), ‖⨆ i, |∑ k, (f i (X (ω k)) - f i (X (x k)))|‖ ∂Measure.pi fun x => μ) ≤
      ∫ (x : Fin n → Ω), ⨆ i, ∑ k, (b + b) ∂Measure.pi fun x => μ)
  (h_original_goal : (∫ (x : Fin n → Ω), ⨆ i, ∑ k, (b + b) ∂Measure.pi fun x => μ) = ↑n * (b + b)) :
  ↑n * b + ↑n * b = ↑n * (b + b) := sorry

State before:
Ω : Type u
inst✝¹ : MeasurableSpace Ω
Z : Type w
n : ℕ
ι : Type u_1
f : ι → Z → ℝ
μ : Measure Ω
inst✝ : IsProbabilityMeasure μ
X : Ω → Z
inst_2 : Nonempty ι
inst_3 : Countable ι
hf : ∀ (i : ι), Measurable (f i ∘ X)
b : ℝ
hf' : ∀ (i : ι) (z : Z), |f i z| ≤ b
ω : Fin n → Ω
this :
  (∫ (x : Fin n → Ω), ‖⨆ i, |∑ k, (f i (X (ω k)) - f i (X (x k)))|‖ ∂Measure.pi fun x => μ) ≤
    ∫ (x : Fin n → Ω), ⨆ i, ∑ k, (b + b) ∂Measure.pi fun x => μ
⊢ (∫ (x : Fin n → Ω), ⨆ i, ∑ k, (b + b) ∂Measure.pi fun x => μ) = ↑n * (b + b)

Tactic: simp only [Finset.sum_const, Finset.card_univ, Fintype.card_fin, smul_add, nsmul_eq_mul, ciSup_const,
  integral_const, measure_univ, ENNReal.one_toReal, smul_eq_mul, one_mul]

State after:
[Ω : Type u
 inst✝¹ : MeasurableSpace Ω
 Z : Type w
 n : ℕ
 ι : Type u_1
 f : ι → Z → ℝ
 μ : Measure Ω
 inst✝ : IsProbabilityMeasure μ
 X : Ω → Z
 inst_2 : Nonempty ι
 inst_3 : Countable ι
 hf : ∀ (i : ι), Measurable (f i ∘ X)
 b : ℝ
 hf' : ∀ (i : ι) (z : Z), |f i z| ≤ b
 ω : Fin n → Ω
 this :
   (∫ (x : Fin n → Ω), ‖⨆ i, |∑ k, (f i (X (ω k)) - f i (X (x k)))|‖ ∂Measure.pi fun x => μ) ≤
     ∫ (x : Fin n → Ω), ⨆ i, ∑ k, (b + b) ∂Measure.pi fun x => μ
 ⊢ ↑n * b + ↑n * b = ↑n * (b + b)]

Relevant lemmas:
smul_eq_mul : ∀ {α : Type u_9} [inst : Mul α] (a b : α), a • b = a * b
MeasureTheory.IsProbabilityMeasure.measure_univ : ∀ {α : Type u_1} {m0 : MeasurableSpace α} {μ : Measure α}
  [self : IsProbabilityMeasure μ], μ Set.univ = 1
MeasureTheory.integral_const : ∀ {α : Type u_1} {E : Type u_2} [inst : NormedAddCommGroup E] [hE : CompleteSpace E]
  [inst_4 : NormedSpace ℝ E] {m : MeasurableSpace α} {μ : Measure α} (c : E), ∫ (x : α), c ∂μ = (μ Set.univ).toReal • c
Finset.card_univ : ∀ {α : Type u_1} [inst : Fintype α], Finset.univ.card = Fintype.card α
Fintype.card_fin : ∀ (n : ℕ), Fintype.card (Fin n) = n
ENNReal.one_toReal : ENNReal.toReal 1 = 1
nsmul_eq_mul : ∀ {α : Type u_1} [inst : NonAssocSemiring α] (n : ℕ) (a : α), n • a = ↑n * a
smul_add : ∀ {M : Type u_1} {A : Type u_7} [inst : AddZeroClass A] [inst_4 : DistribSMul M A] (a : M) (b₁ b₂ : A),
  a • (b₁ + b₂) = a • b₁ + a • b₂
Finset.sum_const : ∀ {α : Type u_3} {β : Type u_4} {s : Finset α} [inst : AddCommMonoid β] (b : β),
  ∑ _x ∈ s, b = s.card • b
one_mul : ∀ {M : Type u} [inst : MulOneClass M] (a : M), 1 * a = a
ciSup_const : ∀ {α : Type u_1} {ι : Sort u_4} [inst : ConditionallyCompleteLattice α] [hι : Nonempty ι] {a : α},
  ⨆ x, a = a