Is there any reference of support for the following theorem?
If a renewal point process with rate $\lambda$ and inter-renewal probability density function $f(x)$ is split into two point processes according to a Bernoulli random variable with parameter $p$, then the resulting point processes are renewal, with rate $\lambda p$ and $\lambda (1-p)$, and with inter-renewal probability function $f_1(x)$ and $f_2(x)$.
[I actually know what $f_1(x)$ and $f_2(x)$ are]
It sounds to me a very well-known theorem, so I am looking for a paper or a book as a reference since