∀ {Ω : Type u_1} {E : Type u_2} {mΩ : MeasurableSpace Ω} {P : MeasureTheory.Measure Ω} [inst : NormedAddCommGroup E]
  [inst_1 : MeasurableSpace E] [BorelSpace E] {X : Ω → E} [inst_3 : NormedSpace ℝ E] [SecondCountableTopology E]
  {Y : Ω → E},
  ProbabilityTheory.HasGaussianLaw (fun ω => (X ω, Y ω)) P → ProbabilityTheory.HasGaussianLaw (fun ω => X ω + Y ω) P