Prove equality (binomial-coefficients)

Question

$$\sum_{i=k}^{n-l} {i \choose k}{n-i \choose l} = {n+1 \choose k+l+1}$$ I understand how to prove this using that equality: $$\sum_{i=k}^{n-l} {i \choose k}{n-i \choose l} = {n \choose k+l} + \sum_{i=k}^{n-l-1} {i \choose k}{n-i-1 \choose l }$$ But I came to it with the help of my intuition, if there is a strict solution, I just want to know about it.