∀ {G : Type u_2} {P : Type u_1} [inst : PartialOrder G] [inst_1 : PartialOrder P] [inst_2 : VAdd G P]
  [inst_3 : IsOrderedCancelVAdd G P] {s : Set G} {t : Set P} {hs : s.IsPWO} {ht : t.IsPWO} (a : P)
  (motive : a ∈ {a | (Finset.VAddAntidiagonal hs ht a).Nonempty} → Prop)
  (x : a ∈ {a | (Finset.VAddAntidiagonal hs ht a).Nonempty}),
  (∀ (b : G × P) (hb : b ∈ Finset.VAddAntidiagonal hs ht a), motive ⋯) → motive x