∀ {R : Type u_1} [inst : LinearOrder R] [inst_1 : CommRing R] [inst_2 : IsStrictOrderedRing R] {S : Type u_3}
  [inst_3 : LinearOrder S] [inst_4 : CommRing S] [IsStrictOrderedRing S] [Archimedean S] (f : S →+*o R) {x : R},
  0 < ArchimedeanClass.mk x → ∀ {y : S}, y < 0 → f y < x