simp only [exists_prop, and_imp, Multiset.prod_cons, Multiset.mem_cons, forall_eq_or_imp]
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)
case intro.refine_2
R : Type u_1
inst✝ : CancelCommMonoidWithZero R
m : Multiset R
⊢ ∀ ⦃a : R⦄ {s : Multiset R},
    (∀ (a b : R),
        a * b ∈ closure {r | IsUnit r ∨ Prime r} →
          (∀ y ∈ s, y ∈ {r | IsUnit r ∨ Prime r}) → Multiset.prod s = a * b → a ∈ closure {r | IsUnit r ∨ Prime r}) →
      ∀ (a_2 b : R),
        a_2 * b ∈ closure {r | IsUnit r ∨ Prime r} →
          a ∈ {r | IsUnit r ∨ Prime r} →
            (∀ y ∈ s, y ∈ {r | IsUnit r ∨ Prime r}) →
              a * Multiset.prod s = a_2 * b → a_2 ∈ closure {r | IsUnit r ∨ Prime r}