∀ {M : Type u_1} [inst : CommMonoid M] {S : Submonoid M} {N : Type u_2} [inst_1 : CommMonoid N] {f : M →* N}
  (h : ∀ (y : ↥S), IsUnit (f ↑y)) {x₁ x₂ : M} {y₁ y₂ : ↥S},
  (f x₁ * ↑((IsUnit.liftRight (f.restrict S) h) y₁)⁻¹ = f x₂ * ↑((IsUnit.liftRight (f.restrict S) h) y₂)⁻¹) =
    (f (x₁ * ↑y₂) = f (x₂ * ↑y₁))