∀ {ι : Type u_1} {σ : Type u_2} {R : Type u_4} [inst : AddCommMonoid ι] [inst_1 : LinearOrder ι] [IsOrderedAddMonoid ι]
  [inst_3 : DecidableEq ι] [CanonicallyOrderedAdd ι] [inst_5 : CommSemiring R] [inst_6 : SetLike σ R]
  [inst_7 : AddSubmonoidClass σ R] {A : ι → σ} [inst_8 : SetLike.GradedMonoid A]
  {s : Multiset (DirectSum ι fun i => ↥(A i))} {m : ι},
  (∀ x ∈ s, ∀ k < m, x k = 0) → ∀ ⦃n : ι⦄, n < s.card • m → s.prod n = 0