induction' f using Multiset.induction_on with p f ih
f : Multiset ℕ
⊢ ∀ {g : Multiset ℕ}, (∀ x ∈ f, OddPrimeOrFour x) → (∀ x ∈ g, OddPrimeOrFour x) → f.prod = g.prod → f = g
case empty

⊢ ∀ {g : Multiset ℕ}, (∀ x ∈ 0, OddPrimeOrFour x) → (∀ x ∈ g, OddPrimeOrFour x) → Multiset.prod 0 = g.prod → 0 = g

case cons
p : ℕ
f : Multiset ℕ
ih : ∀ {g : Multiset ℕ}, (∀ x ∈ f, OddPrimeOrFour x) → (∀ x ∈ g, OddPrimeOrFour x) → f.prod = g.prod → f = g
⊢ ∀ {g : Multiset ℕ},
    (∀ x ∈ p ::ₘ f, OddPrimeOrFour x) → (∀ x ∈ g, OddPrimeOrFour x) → (p ::ₘ f).prod = g.prod → p ::ₘ f = g