We first show that A ∪ (B ∩ C) ⊂ (A ∪ B) ∩ (A ∪ C)

Let x ∈ A ∪ (B ∩ C). Then

x ∈ A or x ∈ B ∩ C

⇒ x ∈ A or (x ∈ B and x ∈ C)

⇒ (x ∈ A or x ∈ B) and (x ∈ A or x ∈ C)

⇒ (x ∈ A ∪ B) and (x ∈ A ∪ C)

⇒ x ∈ (A ∪ B) ∩ (A ∪ C)

Thus,

A ∪ (B ∩ C) ⊂ (A ∪ B) ∩ (A ∪ C) ....(1)

Now we will show that

(A ∪ B) ∩ (A ∪ C) ⊂ (A ∪ C)

Let x ∈ (A ∪ B) ∩ (A ∪ C)

⇒ x ∈ A ∪ B and x ∈ A ∪ C

⇒ (x ∈ A or x ∈ B) and (x ∈ A or x ∈ C)

⇒ x ∈ A or (x ∈ B and x ∈ C)

⇒ x ∈ A or (x ∈ B ∩ C)

⇒ x ∈ A ∪ (B ∩ C)

Thus,

(A ∪ B) ∩ (A ∪ C) ⊂ A ∪ (B ∩ C) ... (2)

So, from (1) and (2), we have

A ∩ (B ∪ C) = (A ∪ B) ∩ (A ∪ C)