∀ {ι : Type u_1} {M : Type u_4} {s : Finset ι} [inst : AddCommMonoid M] {f : ι → M} (g : ι → ι),
  (∀ (a : ι), f a + f (g a) = 0) →
    (∀ (a : ι), f a ≠ 0 → g a ≠ a) → (∀ (a : ι), g a ∈ s) → (∀ (a : ι), g (g a) = a) → ∑ x ∈ s, f x = 0