∀ {ι : Type u_1} {M : Type u_4} [inst : AddCommMonoid M] [inst_1 : PartialOrder M] [IsOrderedCancelAddMonoid M]
  {f g : ι → M} {s : Finset ι}, (∀ i ∈ s, f i ≤ g i) → (∃ i ∈ s, f i < g i) → ∑ i ∈ s, f i < ∑ i ∈ s, g i