$[B/\mathfrak{P}^n:A/p]=\sum_{k=0}^{n-1}[\mathfrak{P}^k/\mathfrak{P}^{k+1}:A/p]=n[B/\mathfrak{P}:A/p]$

Question

If $A$ is a Dedekind domain with field of quotients $K$, $L$ is a finite separable extension of $K$ and $B$ is the integral closure of $A$ in $L$ then it is known that $B$ is also a Dedekind domain. Now if $p$ is a maximal ideal of $A$ and $\mathfrak{P}$ is a maximal ideal of $B$ such that $\mathfrak{P}\cap A=p$ then how exactly do we prove the following relation :
$[B/\mathfrak{P}^n:A/p]=\sum_{k=0}^{n-1}[\mathfrak{P}^k/\mathfrak{P}^{k+1}:A/p]=n[B/\mathfrak{P}:A/p]$ where $n=e_{\mathfrak{P}}$.
I think that I managed to prove the first equality :
$[\mathfrak{P}^k/\mathfrak{P}^{k+1}:A/p]=[(\mathfrak{P}^k/\mathfrak{P}^n)/(\mathfrak{P}^{k+1}/\mathfrak{P}^n):A/p]=[\mathfrak{P}^k/\mathfrak{P}^{n}:A/p]-[\mathfrak{P}^{k+1}/\mathfrak{P}^{n}:A/p]$
So $\sum_{k=0}^{n-1}[\mathfrak{P}^k/\mathfrak{P}^{k+1}:A/p]=[B/\mathfrak{P}^n:A/p]$.
Is this correct ?
As for the second equality, it seems that I have to prove that $[\mathfrak{P}^k/\mathfrak{P}^{k+1}:A/p]=[B/\mathfrak{P}:A/p]\ \forall 0\le k<n$. I don't know how to prove it though.
Any help would be appreciated.