In the following problem:
If the space $S$ is a set of positive numbers, How to show that if $P\{t_0 \leq t \leq t_0 + t_1 | t \geq t_0\} = P\{t \leq t_1\}$ for all $t_0$ and $t_1$ then $P\{t\leq t_1\}=1-e^{-ct_1}$.
I don't get how he moves from:
$ \frac{\int_{t_0}^{t_0 + t_1}{\alpha(t)\mathrm{d}t}}{\int_{t_0}^{+\infty}{\alpha(t)\mathrm{d}t}} = \int_{0}^{t_1}{\alpha(t)\mathrm{d}t}$
to the following: $ \frac{\alpha(t_0)}{\int_{t_0}^{+\infty}{\alpha(t)\mathrm{d}t}} = \alpha(0).$