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