Proving $A \cup (B \cap C) = (A \cup B) \cap (A \cup C)$.

You must first prove 2 cases:

(1) $A \cap (B \cup C) \subset (A \cap B) \cup (A \cap C)$

(2) $(A \cap B) \cup (A \cap C) \subset A \cap (B \cup C)$

Note that in mathematics we use the following symbols:

$\cap=$ AND = $\land$

$\cup=$ OR = $\lor$

Let $x \in A \cap (B \cup C) \implies x \in A \land x \in (B \cup C)$

$\implies x \in A \land \{ x \in B \lor x \in C \}$

$\implies \{ x \in A \land x \in B \} \lor\{ x \in A \land x \in C \} $

$\implies x \in (A \cap B) \lor x \in (A \cap C)$

$\implies x \in (A \cap B) \cup (A \cap C)$

$\therefore x \in A \cap (B \cup C) \implies x \in (A \cap B) \cup (A \cap C)$

$\therefore A \cap (B \cup C) \subset (A \cap B) \cup (A \cap C)$

Let $x \in (A \cap B) \cup (A \cap C) \implies x \in (A \cap B) \lor x \in (A \cap C)$

$\implies \{x \in A \land x \in B \} \lor \{ x \in A \land x \in C \}$

$\implies x \in A \land \{ x \in B \lor x \in C\}$

$\implies x \in A \land \{B \cup C \}$

$\implies x \in A \cap (B \cup C)$

$\therefore x \in (A \cap B) \cup (A \cap C) \implies x \in A \cap (B \cup C)$

$\therefore (A \cap B) \cup (A \cap C) \subset A \cap (B \cup C)$

$\therefore A \cup (B \cap C) = (A \cup B) \cap (A \cup C)$