Let $f$ be an integrable function on [0,1]. Prove that on any interval $J\subset[0,1]$ contained in some $I_{n,l}$ we have $$\frac{1}{|J|}\int_{J}f(x)dx = \sum_{j = 0}^{n} \sum_{k=0}^{2^{j} - 1}<f,h_{j,k}>h_{j,k}(x).$$
Could anyone give me a hint please?