∀ {M : Type u_1} {N : Type u_2} [inst : Monoid M] [inst_1 : Monoid N] {x z : M × N}
  (motive : Associated x.1 z.1 ∧ Associated x.2 z.2 → Prop) (x_1 : Associated x.1 z.1 ∧ Associated x.2 z.2),
  (∀ (u₁ : Mˣ) (h₁ : x.1 * ↑u₁ = z.1) (u₂ : Nˣ) (h₂ : x.2 * ↑u₂ = z.2), motive ⋯) → motive x_1