How to Solve Disjunction Elimination Proof

Question

How would you prove the following by disjunction elimination?

Premises: A ∨ B , A ∨ C

Conclusion: A ∨ (B ∧ C)

Answer 1

1. Premise: A v B
2. Case 1: A
3. Then, we can just get A v (B ∧ C) and it's works.
4. Case 2: B (where A is false)
5. Premise: A v C
6. since NOT A, we get C (because A v C is true)
7. so, A v (B ∧ C)