∀ {M : Type u_1} [inst : Monoid M] (S : Submonoid M) {x : Mˣ},
  x ∈ (Submonoid.comap (Units.coeHom M) S ⊓ (Submonoid.comap (Units.coeHom M) S)⁻¹).carrier →
    x⁻¹ ∈ ↑(Submonoid.comap (Units.coeHom M) S) ∧ x⁻¹ ∈ ↑(Submonoid.comap (Units.coeHom M) S)⁻¹