∀ {M : Type u_1} [inst : AddMonoid M] (S : AddSubmonoid M) (x : AddUnits ↥S),
  { val := ↑↑x, neg := ↑↑(-x), val_neg := ⋯, neg_val := ⋯ } ∈ ↑(AddSubmonoid.comap (AddUnits.coeHom M) S) ∧
    { val := ↑↑x, neg := ↑↑(-x), val_neg := ⋯, neg_val := ⋯ } ∈ ↑(-AddSubmonoid.comap (AddUnits.coeHom M) S)