rcases Set.nonempty_iff_ne_empty.2 (ha₂ hP) with ⟨x, ⟨hx, hx₂⟩⟩
R : Type u_1
inst✝¹ : CommSemiring R
inst✝ : IsDomain R
H : ∀ (I : Ideal R), I ≠ ⊥ → Ideal.IsPrime I → ∃ x ∈ I, Prime x
hP : Set.Nonempty {r | Prime r}
a : R
ha : a ≠ 0
ha₂ : Set.Nonempty {r | Prime r} → ↑(Ideal.span {a}) ∩ ↑(Submonoid.closure {r | Prime r}).toSubsemigroup ≠ ∅
⊢ ∃ f, (∀ b ∈ f, Prime b) ∧ Associated (Multiset.prod f) a
case intro.intro
R : Type u_1
inst✝¹ : CommSemiring R
inst✝ : IsDomain R
H : ∀ (I : Ideal R), I ≠ ⊥ → Ideal.IsPrime I → ∃ x ∈ I, Prime x
hP : Set.Nonempty {r | Prime r}
a : R
ha : a ≠ 0
ha₂ : Set.Nonempty {r | Prime r} → ↑(Ideal.span {a}) ∩ ↑(Submonoid.closure {r | Prime r}).toSubsemigroup ≠ ∅
x : R
hx : x ∈ ↑(Ideal.span {a})
hx₂ : x ∈ ↑(Submonoid.closure {r | Prime r}).toSubsemigroup
⊢ ∃ f, (∀ b ∈ f, Prime b) ∧ Associated (Multiset.prod f) a