∀ {M : Type u_1} [inst : AddCommGroup M] [inst_1 : LinearOrder M] [inst_2 : IsOrderedAddMonoid M] {a : M}
  {s : UpperSet (ArchimedeanClass M)}, s ≠ ⊤ → (a ∈ ArchimedeanClass.addSubgroup s ↔ ArchimedeanClass.mk a ∈ s)