∀ {ι : Type u_1} {α : ι → Type u_6} [Finite ι] {r : (i : ι) → α i → α i → Prop} [∀ (i : ι), IsPreorder (α i) (r i)]
  {s : (i : ι) → Set (α i)},
  (∀ (i : ι), (s i).PartiallyWellOrderedOn (r i)) →
    (Set.univ.pi s).PartiallyWellOrderedOn fun a b => ∀ (i : ι), r i (a i) (b i)