∀ {M : Type u_1} [inst : AddCommMonoid M] (S : AddSubmonoid M) {x y : M × ↥S}
  (motive : (∃ c, ↑c + (↑y.2 + x.1) = ↑c + (↑x.2 + y.1)) → Prop) (x_1 : ∃ c, ↑c + (↑y.2 + x.1) = ↑c + (↑x.2 + y.1)),
  (∀ (c : ↥S) (hc : ↑c + (↑y.2 + x.1) = ↑c + (↑x.2 + y.1)), motive ⋯) → motive x_1