Show the independence property for Standard Homogenous Poisson Process

Question

Problem:

Let $\tilde{N}$ be a Standard Homogenous Poisson Proces on $[0,\infty)$, and let N be a Poisson process on $[0,\infty)$ with mean value function $\mu$

Show that $N=(\tilde{N}(\mu(t)))_{t\geq0}$ is a Poisson Process on $[0,\infty)$ with mean value function $\mu$.

Attempt: The definition says:

1. The process starts at 0: $N(0)=0 \: a.s.$

2.The process has independent increments:

For any $t_i , \: i=0,1,...,n$ and $n\geq 1$ such that $0=t_0<t_1<...<t_n ,$ the increments $N(t_{i-1},t_i], \: i=1,...,n$ are mutually independent.

3. $N(s,t]\sim Pois(\mu(s,t])$

4. N has the cádlág property.

I can do every one of them except the 2. increment property. How do I start on it? What is the game plan?