$\begin{align}  & \text{From the L}\text{.H}\text{.S} \\ & \text{Let }a\in (X\cup Y)' \\ & a\notin X\cup Y \\ & a\notin X\text{ and }a\notin Y\text{ } \\ & a\in X'\text{ and }a\in Y' \\ & a\in X'\cap Y' \\ & X'\cap Y'\subseteq (X\cup Y)'---(i) \\ & \text{From R}\text{.H}\text{.S} \\ & \text{Let }b\in X'\cap Y' \\ & b\in X'\text{ and }b\in Y' \\ & b\notin X\text{ or }b\notin Y \\ & b\notin (X\cup Y) \\ & b\in (X\cup Y)' \\ & (X\cup Y)'\subseteq X'\cap Y'---(ii) \\ & \therefore (X\cup Y)'=X'\cap Y' \\\end{align}$