classical
ext P
have : p.map (algebraMap R S) ≠ ⊥ := by
  rwa [Ne, Ideal.map_eq_bot_iff_of_injective (NoZeroSMulDivisors.algebraMap_injective R S)]
simp only [Finset.mem_coe, Multiset.mem_toFinset, Ideal.mem_normalizedFactors_iff this,
  factors_eq_normalizedFactors, Ideal.map_le_iff_le_comap, primesOverFinset]
exact ⟨fun H ↦ ⟨H.1, ((hp'.isMaximal hp).eq_of_le (H.1.comap _).ne_top H.2).symm⟩,
  fun H ↦ ⟨H.1, H.2.ge⟩⟩
R : Type u_1
S : Type u_2
inst✝⁵ : CommRing R
inst✝⁴ : CommRing S
inst✝³ : Algebra R S
inst✝² : Ring.DimensionLEOne R
inst✝¹ : IsDedekindDomain S
inst✝ : NoZeroSMulDivisors R S
p : Ideal R
hp : p ≠ ⊥
hp' : p.IsPrime
⊢ ↑(primesOverFinset S p) = primesOver S p
no goals