Step 1: Start with the Left Hand Side (LHS). $\text{LHS} = \overline{A \cup (B \cap C)}$

Step 2: Apply De Morgan's Law: $\overline{X \cup Y} = \overline{X} \cap \overline{Y}$. Let $X = A$ and $Y = (B \cap C)$. $\text{LHS} = \overline{A} \cap \overline{(B \cap C)}$

Step 3: Apply De Morgan's Law again on the second term: $\overline{X \cap Y} = \overline{X} \cup \overline{Y}$. $\overline{(B \cap C)} = \overline{B} \cup \overline{C}$ Substitute this back into the LHS equation: $\text{LHS} = \overline{A} \cap (\overline{B} \cup \overline{C})$

Step 4: Use the Commutative Law of Union: $X \cup Y = Y \cup X$. $\overline{B} \cup \overline{C} = \overline{C} \cup \overline{B}$ $\text{LHS} = \overline{A} \cap (\overline{C} \cup \overline{B})$

Step 5: Use the Commutative Law of Intersection: $X \cap Y = Y \cap X$. $\text{LHS} = (\overline{C} \cup \overline{B}) \cap \overline{A}$

Conclusion: $\text{LHS} = \text{RHS}$ Hence, proved.