∀ {R : Type u_1} {S : Type u_2} [inst : CommRing R] [inst_1 : IsLocalRing R] [inst_2 : CommRing S]
  [inst_3 : IsLocalRing S] [inst_4 : Algebra R S] [inst_5 : IsLocalHom (algebraMap R S)] (x : R),
  (algebraMap (IsLocalRing.ResidueField R) (IsLocalRing.ResidueField S)) ((IsLocalRing.residue R) x) =
    (IsLocalRing.residue S) ((algebraMap R S) x)