∀ {α : Type u_5} {f : Filter α} {R : Type u_6} {κ : Type u_7} [inst : AddCommMonoid R] {r : R → R → Prop},
  (∀ (v₁ v₂ : α → R),
      Filter.IsBoundedUnder r f v₁ → Filter.IsBoundedUnder r f v₂ → Filter.IsBoundedUnder r f (v₁ + v₂)) →
    r 0 0 →
      ∀ {u : κ → α → R} (s : Finset κ),
        (∀ k ∈ s, Filter.IsBoundedUnder r f (u k)) → Filter.IsBoundedUnder r f (∑ k ∈ s, u k)