Suppose the real random variables $X$ and $Y$ are independent; i.e. $\mathbb{P}[\{X \leq x\} \cap \{Y \leq y\}] = \mathbb{P}[\{X \leq x\}] \cdot \mathbb{P}[\{Y \leq y\}].$
Does it follow that $\mathbb{P}[\{X > x\} \cap \{Y > y\}] = \mathbb{P}[\{X > x\}] \cdot \mathbb{P}[\{Y > y\}]?$