∀ {M : Type u_1} [inst : CommGroup M] [inst_1 : LinearOrder M] [inst_2 : IsOrderedMonoid M] {a b : M},
  (MulArchimedeanClass.mk a = MulArchimedeanClass.mk b) = ((∃ m, |b|ₘ ≤ |a|ₘ ^ m) ∧ ∃ n, |a|ₘ ≤ |b|ₘ ^ n)