Use the order statistics property to show that for $s<1 \text{ and } t\leq n $ that $P(N(s)=k|N(1)=n)={{n}\choose{k}} s^k (1-s)^{n-k}$

Question

Problem:

Use the order statistics property to show that for $s<1 \text{ and } k\leq n,$

$$P(N(s)=k|N(1)=n)={{n}\choose{k}} s^k (1-s)^{n-k}$$

Theorem: Consider the Poisson process N=(N(t)), $t\geq 0$, with continous a.e. positive intensity function $\lambda$ and arrival times $0<T_1<T_2<... \text{ a.s.}.$ Then the conditional distribution of $(T_1,...,T_n)$ given $\{N(t)=n\}$ is the distribution of the ordered sample $(X_{(1)},...,X_{(n)})$ of an iid sample $X_1 ,..., X_n$ with common density $\frac{\lambda(x)}{\mu(t)}, \: 0<x\leq t:$

$$(T_1 ,..., T_n | N(t)=n)\stackrel{d}{=}(X_{(1)},...,X_{(n)})$$

In other words, the left-hand vector has conditional density

$$f_{T_1 , ... , T_n}(x_1 ,..., x_n | N(t)=n)=\frac{n!}{(\mu(t))^n}\displaystyle\prod_{i=1}^n \lambda(x_i), \: 0< x_1 < ... <x_n <t.$$

Proposition Let $(X_i)$ be a iid sequence, independent of the sequence $(T_i)$ of arrival times of a homogeneous Poisson process $N$ with intensity $\lambda.$ Then for any measurable function $g: \mathbb{R}^2 \to \mathbb{R}$ the following idenity in distribution holds: $$S(t)=\displaystyle\sum_{i=1}^{N(t)}g(T_i ,X_i)\stackrel{d}{=}\displaystyle\sum_{i=1}^{N(t)}g(tU_i , X_i),$$ where $(U_i)\stackrel{iid}{\sim}U(0,1)$ sequence, independent of $(X_i)$ and $(T_i).$

Solution to Problem:

Applying the order statistics property of the homogeneous Poisson process we have for the arrivals $T_1 < T_2 <...$ $$(T_1 , ..., T_n | N(1)=n)\stackrel{d}{=}(U_{(1)}, ..., U_{(n)}),$$ where $U_{(1)}<...<U_{(n)}$ are the order statistics of an iid $U(0,1)$ sample $U_1 ,..., U_n .$ Then for $s<1, \: k\leq n :$

$$ \begin{aligned} P(N(s)=k|N(1)=n) &= P(\sum_{i=1}^{n}\mathbb{1}_{(T_i \leq s)}=k|N(1)=n) \\ &= P(\sum_{i=1}^{n}\mathbb{1}_{(U_{(i)} \leq s)}=k|N(1)=n) \\ &=P(\sum_{i=1}^{n}\mathbb{1}_{(U_i \leq s)}=k|N(1)=n) \\ &= {{n}\choose{k}} s^k (1-s)^{n-k} \end{aligned} $$

Question:

1. I don't understand how to use the Theorem nor the Proposition.
2. I've tried and stared myself blind at problem... I can't understand any of the equality signs...

Can anyone please either guide me through it or drop me some hints?

Thanks in advance.