∀ {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 : InnerProductSpace ℝ E] [inst_11 : InnerProductSpace ℝ F]
  {X : Ω → E} {Y : Ω → F},
  ProbabilityTheory.HasGaussianLaw (fun ω => (X ω, Y ω)) P →
    (∀ (x : E) (y : F), ProbabilityTheory.covariance (fun ω => inner ℝ x (X ω)) (fun ω => inner ℝ y (Y ω)) P = 0) →
      ProbabilityTheory.IndepFun X Y P