simp only [Set.mem_setOf_eq, forall_apply_eq_imp_iff, Multiset.mem_cons, forall_eq_or_imp,
  Multiset.prod_cons, and_imp, forall_apply_eq_imp_iff₂]
case intro.intro.refine'_2
R : Type u_1
inst✝ : CommMonoidWithZero R
l : Multiset R
⊢ ∀ ⦃a : R⦄ {s : Multiset R},
    (fun l =>
          ∀ (a : R),
            (∀ y ∈ l, y ∈ {r | IsUnit r ∨ Prime r}) →
              Multiset.prod l = a → ∃ f, (∀ b ∈ f, Prime b) ∧ Associated (Multiset.prod f) a)
        s →
      (fun l =>
          ∀ (a : R),
            (∀ y ∈ l, y ∈ {r | IsUnit r ∨ Prime r}) →
              Multiset.prod l = a → ∃ f, (∀ b ∈ f, Prime b) ∧ Associated (Multiset.prod f) a)
        (a ::ₘ s)
case intro.intro.refine'_2
R : Type u_1
inst✝ : CommMonoidWithZero R
l : Multiset R
⊢ ∀ ⦃a : R⦄ {s : Multiset R},
    ((∀ y ∈ s, IsUnit y ∨ Prime y) → ∃ f, (∀ b ∈ f, Prime b) ∧ Associated (Multiset.prod f) (Multiset.prod s)) →
      IsUnit a ∨ Prime a →
        (∀ y ∈ s, IsUnit y ∨ Prime y) → ∃ f, (∀ b ∈ f, Prime b) ∧ Associated (Multiset.prod f) (a * Multiset.prod s)