refine m.induction (p := fun m ↦ ∀ a b, a * b ∈ closure {r | IsUnit r ∨ Prime r} →
  (∃ (_ : ∀ y ∈ m, y ∈ {r | IsUnit r ∨ Prime r}), m.prod = a * b) →
  a ∈ closure {r | IsUnit r ∨ Prime r}) (fun _ _ _ hprod => subset_closure (Set.mem_def.2 ?_)) ?_
case intro
R : Type u_1
inst✝ : CancelCommMonoidWithZero R
m : Multiset R
⊢ ∀ (a b : R),
    a * b ∈ closure {r | IsUnit r ∨ Prime r} →
      (∃ (_ : ∀ y ∈ m, y ∈ {r | IsUnit r ∨ Prime r}), Multiset.prod m = a * b) → a ∈ closure {r | IsUnit r ∨ Prime r}
case intro.refine_1
R : Type u_1
inst✝ : CancelCommMonoidWithZero R
m : Multiset R
x✝² x✝¹ : R
x✝ : x✝² * x✝¹ ∈ closure {r | IsUnit r ∨ Prime r}
hprod : ∃ (_ : ∀ y ∈ 0, y ∈ {r | IsUnit r ∨ Prime r}), Multiset.prod 0 = x✝² * x✝¹
⊢ {r | IsUnit r ∨ Prime r} x✝²

case intro.refine_2
R : Type u_1
inst✝ : CancelCommMonoidWithZero R
m : Multiset R
⊢ ∀ ⦃a : R⦄ {s : Multiset R},
    (fun m =>
          ∀ (a b : R),
            a * b ∈ closure {r | IsUnit r ∨ Prime r} →
              (∃ (_ : ∀ y ∈ m, y ∈ {r | IsUnit r ∨ Prime r}), Multiset.prod m = a * b) →
                a ∈ closure {r | IsUnit r ∨ Prime r})
        s →
      (fun m =>
          ∀ (a b : R),
            a * b ∈ closure {r | IsUnit r ∨ Prime r} →
              (∃ (_ : ∀ y ∈ m, y ∈ {r | IsUnit r ∨ Prime r}), Multiset.prod m = a * b) →
                a ∈ closure {r | IsUnit r ∨ Prime r})
        (a ::ₘ s)