Distributive Laws/Set Theory/Examples/A cap B cap (C cup D) subset of (A cap D) cup (B cap C)

Theorem
Let:
 * $P = A \cap B \cap \paren {C \cup D}$
 * $Q = \paren {A \cap D} \cup \paren {B \cap C}$

Then:
 * $P \subseteq Q$

Proof
Let $P$ be expressed as:
 * $P = X \cup Y$

where:
 * $X = A \cap B \cap D$
 * $Y = A \cap B \cap C$

Then:

and:

and so: