∀ {Ω : Type u_1} {ι : Type u_4} {mΩ : MeasurableSpace Ω} {P : MeasureTheory.Measure Ω} [inst : Fintype ι] {E : Type u_6}
  [inst_1 : NormedAddCommGroup E] [inst_2 : NormedSpace ℝ E] [inst_3 : MeasurableSpace E] [BorelSpace E]
  [SecondCountableTopology E] {X : ι → Ω → E},
  ProbabilityTheory.HasGaussianLaw (fun ω x => X x ω) P → ProbabilityTheory.HasGaussianLaw (fun ω => ∑ i, X i ω) P