∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {P : MeasureTheory.Measure Ω} {E : Type u_2} {F : Type u_3}
  [inst : NormedAddCommGroup E] [inst_1 : MeasurableSpace E] [CompleteSpace E] [BorelSpace E]
  [SecondCountableTopology E] [inst_5 : NormedAddCommGroup F] [inst_6 : MeasurableSpace F] [CompleteSpace F]
  [BorelSpace F] [SecondCountableTopology F] [inst_10 : NormedSpace ℝ E] [inst_11 : NormedSpace ℝ F] {X : Ω → E}
  {Y : Ω → F},
  ProbabilityTheory.HasGaussianLaw (fun ω => (X ω, Y ω)) P →
    (∀ (L₁ : StrongDual ℝ E) (L₂ : StrongDual ℝ F), ProbabilityTheory.covariance (⇑L₁ ∘ X) (⇑L₂ ∘ Y) P = 0) →
      ProbabilityTheory.IndepFun X Y P