$M/G/\infty$ queue arrival and departure distributions

Question

Suppose we have an $M/G/\infty$ queue. CDF is $F(.)$

Question: Let $Z(t)$ be number of customers in system at time $t$.

1. Let $F(t)$ represent the number of customers that have departed system by time t. Find $E[F(t)]$

Lalley presents a beautiful proof of $E[Z(t)]$:

So he gets that $Z(t)$ is Poisson with mean $\int_{0}^{t}\int_{t-s}^{\infty}\lambda f(y)=\lambda \int_{0}^{t}(1-F(t-s))ds=\lambda\int_{0}^{t} 1-F(y)dy$ ,where $y =t-s$

Now I believe that $E[F(t)]=\lambda \int_{0}^{t} F(t-s)ds=\int_{0}^{t} F(y)dy$. Ross proves this using Type 1 and Type 2 customers etc. I was wondering how I would prove this using in a similar manner to the above proof using a two-dimensional point Process. I feel like it can be done and the proof will be very similar although I'm not exactly sure and I'm not exactly sure how to set it up/set the integral up.