theorem extracted_formal_statement_290.{u_3, u_2, u_1} {Ω : Type u_1} [inst : MeasurableSpace Ω] {μ : Measure Ω}
  [inst_1 : IsProbabilityMeasure μ] {𝓧 : Type u_2} {ι : Type u_3} {m𝓧 : MeasurableSpace 𝓧} [inst_2 : Fintype ι]
  {X' : Ω → 𝓧} (hX'' : Measurable X') (S T : Finset ι) (hST : Disjoint T S) :
  IndepFun (fun ω t => X' (ω ↑t)) (fun ω s => X' (ω ↑s)) (Measure.pi fun x => μ) := sorry

State before:
Ω : Type u_1
inst✝³ : MeasurableSpace Ω
μ : Measure Ω
inst✝² : IsProbabilityMeasure μ
𝓧 : Type u_2
ι : Type u_3
inst✝¹ : DecidableEq ι
m𝓧 : MeasurableSpace 𝓧
inst✝ : Fintype ι
X' : Ω → 𝓧
hX'' : Measurable X'
S T : Finset ι
hST : Disjoint T S
⊢ IndepFun (fun ω t => X' (ω ↑t)) (fun ω s => X' (ω ↑s)) (Measure.pi fun x => μ)

Tactic: exact iIndepFun.indepFun_finset T S hST (pi_comp_eval_iIndepFun hX'') fun _ ↦ (by measurability)

State after:

Relevant lemmas:
pi_comp_eval_iIndepFun : ∀ {Ω : Type u_1} {ι : Type u_2} [inst : MeasurableSpace Ω] {μ : Measure Ω}
  [inst_3 : IsProbabilityMeasure μ] [inst_4 : Fintype ι] [inst_5 : DecidableEq ι] {𝓧 : Type u_3}
  [inst_6 : MeasurableSpace 𝓧] {X : Ω → 𝓧},
  Measurable X → iIndepFun (fun i => X ∘ Function.eval i) (Measure.pi fun x => μ)
ProbabilityTheory.iIndepFun.indepFun_finset : ∀ {Ω : Type u_1} {ι : Type u_2} {_mΩ : MeasurableSpace Ω} {μ : Measure Ω}
  {β : ι → Type u_10} {m : (i : ι) → MeasurableSpace (β i)} {f : (i : ι) → Ω → β i} (S T : Finset ι),
  Disjoint S T → iIndepFun f μ → (∀ (i : ι), Measurable (f i)) → IndepFun (fun a i => f (↑i) a) (fun a i => f (↑i) a) μ