∀ (s : Set ℕ) (motive : (∃ t, ↑t ⊆ s ∧ ∃ c, ∑ i, c i • ↑↑i = ↑(Nat.setGcd s)) → Prop)
  (x : ∃ t, ↑t ⊆ s ∧ ∃ c, ∑ i, c i • ↑↑i = ↑(Nat.setGcd s)),
  (∀ (t : Finset ℕ) (hts : ↑t ⊆ s) (a : ↥t → ℤ) (eq : ∑ i, a i • ↑↑i = ↑(Nat.setGcd s)), motive ⋯) → motive x