Prove `A cup (B cap C)=(A cup B) cap(A cup C)`

Explanation

To prove the set equality \( A \cup (B \cap C) = (A \cup B) \cap (A \cup C) \), we will show that each side is a subset of the other.

### Step 1: Show that \( A \cup (B \cap C) \subseteq (A \cup B) \cap (A \cup C) \)

1. Let \( x \in A \cup (B \cap C) \).
2. This means that \( x \) is either in \( A \) or in \( B \cap C \).
3. If \( x \in A \), then \( x \in A \cup B \) and \( x \in A \cup C \). Thus, \( x \in (A \cup B) \cap (A \cup C) \).
4. If \( x \in B \cap C \), then \( x \in B \) and \( x \in C \).
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