∀ {Ω : Type u_1} {E : Type u_2} {F : Type u_3} {mΩ : MeasurableSpace Ω} {P : MeasureTheory.Measure Ω}
  [inst : NormedAddCommGroup E] [inst_1 : MeasurableSpace E] [BorelSpace E] {X : Ω → E} [inst_3 : NormedSpace ℝ E]
  [inst_4 : NormedAddCommGroup F] [inst_5 : NormedSpace ℝ F] [inst_6 : MeasurableSpace F] [BorelSpace F] {Y : Ω → F}
  [SecondCountableTopologyEither E F] (p : ENNReal) [inst_9 : Fact (1 ≤ p)],
  ProbabilityTheory.HasGaussianLaw (fun ω => (X ω, Y ω)) P →
    ProbabilityTheory.HasGaussianLaw (fun ω => WithLp.toLp p (X ω, Y ω)) P