∀ {G : Type u_1} [inst : AddGroup G] {ι : Type u_2} {H : ι → AddSubgroup G} {s : Finset ι}
  [inst_1 : DecidableEq (AddSubgroup G)] (motive : (∃ a, a = (Finset.image H s).card) → Prop)
  (x : ∃ a, a = (Finset.image H s).card), (∀ (n : ℕ) (hn : n = (Finset.image H s).card), motive ⋯) → motive x