Let $\{X\_t\}\_{t\ge 0}$ be a Poisson Process with parameter $\lambda$. Suppose that each event is type 1 with probability $\alpha$ and type 2 with probability $1-\alpha$. Let $\{X^{(1)}\_t\}\_{t\ge 0}$ the number of type 1 events up until time $t$ and $\{X^{(2)}\_t\}\_{t\ge 0}$ the number of type 2 events up until time $t$ 

Prove that $\{X^{(1)}\_t\}\_{t\ge 0}$ and $\{X^{(2)}\_t\}\_{t\ge 0}$ are Poisson Processes with parameter $\lambda \alpha$ and $\lambda(1-\alpha)$ respectively 

Furrthermore prove that for each $t\ge 0$ the random variables $\{X^{(1)}\_t\}\_{t\ge 0}$ and $\{X^{(2)}\_t\}\_{t\ge 0}$ are independent

My attempt: In order to prove that they are poisson process I will use the next definition:

An stochastic process $\{Y\_t\}\_{t\ge 0}$ is a poisson process iff:

a) $Y\_0=0$

b) It has independent increments

c) $Y\_{t+s}-Y\_{s}$~$Poisson(\lambda t)$ for any values $s\ge 0$ and $t>0$

a) For any $t\ge 0$ we have: $X\_t=X^{(1)}\_t+X^{(2)}\_t$; we know that $\{X\_t\}\_{t\ge 0}$ is a poisson process hence $X\_0=0$ $\Rightarrow X^{(1)}\_0+X^{(2)}\_0=0 \Rightarrow X^{(1)}\_0=0$ and $X^{(2)}\_0=0$

b)Let $n\in \mathbb N$ In this part I need to prove that for any $n$ arbitrary times $0<t\_1\le t\_2\le...\le t\_n$ and states $x\_1,...,x\_n$
 $$P[X^{(1)}\_{t\_1}=x\_1,X^{(1)}\_{t\_2}-X^{(1)}\_{t\_1}=x\_2,...,X^{(1)}\_{t\_n}-X^{(1)}\_{t\_{n-1}}=x\_n]=P[X^{(1)}\_{t\_1}=x\_1]P[X^{(1)}\_{t\_2}-X^{(1)}\_{t\_1}=x\_2]...P[X^{(1)}\_{t\_n}-X^{(1)}\_{t\_{n-1}}=x\_n]$$

I don´t know how to Formally prove this part, and I don´t think this is trivial. Any help would be highly appreciated 

c) $$P[X^{(1)}\_t=k]=\sum\_{i=k}^\infty P[X^{(1)}\_t=k|X\_t=i]P[X\_t=i]=\sum\_{i=k}^\infty \binom{i}{k}\alpha^i(1-\alpha)^{i-k}{e^{-\lambda t}(\lambda t)^i \over i!}={e^{-\lambda \alpha t}(\lambda \alpha t)^k\over k!}$$

Know I need to compute $P[X^{(1)}\_{t+s}-X^{(1)}\_t=n]=\sum\_{j=0}^\infty P[X^{(1)}\_{t+s}-X^{(1)}\_t=n|X^{(1)}\_s=j]P[X^{(1)}\_s=j]$

This part is also giving me trouble because I dont´know what to do from here 

I would really apreciate if you can help me with this problem. Also I hope that this won´t be marked as a duplicate because I haven´t seen a formal proof about the splitting poisson process.