In this paper it is shown that for a minimal system $(X,T)$ and $d,k\in \mathbb {N}$ , if $(x,x_{i})$ is regionally proximal of order d for $1\leq i\leq k$ , then $(x,x_{1},\ldots ,x_{k})$ is $(k+1)$ -regionally proximal of order d. Meanwhile, we introduce the notion of $\mathrm {IN}^{[d]}$ -pair: for a dynamical system $(X,T)$ and $d\in \mathbb {N}$ , a pair $(x_{0},x_{1})\in X\times X$ is called an $\mathrm {IN}^{[d]}$ -pair if for any $k\in \mathbb {N}$ and any neighborhoods $U_{0} ,U_{1} $ of $x_{0}$ and $x_{1}$ respectively, there exist different $(p_{1}^{(i)},\ldots ,p_{d}^{(i)})\in \mathbb {N}^{d} , 1\leq i\leq k$ , such that

where $\mathrm {Ind}(U_{0},U_{1})$ denotes the collection of all independence sets for $(U_{0},U_{1})$ . It turns out that for a minimal system, if it does not contain any non-trivial $\mathrm {IN}^{[d]}$ -pair, then it is an almost one-to-one extension of its maximal factor of order d.