Prove that $\sum_{k=0}^{n-1}\frac{1}{\prod_{m\neq k}(a_m-a_k)}=0$ for any sequence $a_n$.

Question

Let $n>1$ be natural number and $a_0,a_1,\dots,a_{n-1}$ be the sequence of parwise different real numbers. Also let $A_k=\{0,1,\dots,n-1\}\setminus \{k\}$. I'm looking for a simple proof of following identity: $$\sum_{k=0}^{n-1}\frac{1}{\prod_{m\in A_k}(a_m-a_k)}=0$$ I've been able to show this by using induction over $n$, reducing the sum to common denominator and then considering numerator as polynomial in $a_{n-1}$ and showing that it is equal to $0$ in $n$ points and it's degree is at most $n-1$. However I believe there has to be some more elementary proof of this identity.
So my questions are: is there simpler proof of this identity, are there similar identities known in the literature?