(Lack of memory of the exponential distribution) Let $X>0$ be a strictly positive random variable and let $\theta>0 .$ Show that $X$ is exponentially distributed if and only if
$$
\mathbf{P}[X>t+s \mid X>s]=\mathbf{P}[X>t] \quad \text { for all } s, t \geq 0
$$
In particular, $X \sim \exp _{\theta}$ if and only if $\mathbf{P}[X>t+s \mid X>s]=e^{-\theta t}$ for all $s, t \geq 0$.