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By proving the minimality of face transformations acting on the diagonalpoints and searching the points allowed in the minimal sets, it is shown thatthe regionally proximal relation of order $d$, $\RP^{[d]}$, is an equivalencerelation for minimal systems. Moreover, the lifting of $\RP^{[d]}$ between twominimal systems is obtained, which implies that the factor induced by$\RP^{[d]}$ is the maximal $d$-step nilfactor. The above results extend thesame conclusions proved by Host, Kra and Maass for minimal distal systems.  A combinatorial consequence is that if $S$ is a dynamically syndetic subsetof $\Z$, then for each $d\ge 1$, $$\{(n_1,\...,n_d)\in \Z^d: n_1\ep_1+...+n_d\ep_d\in S, \ep_i\in \{0,1\}, 1\le i\le d\}$$ is syndetic. In some sensethis is the topological correspondence of the result obtained by Host and Krafor positive upper Banach density subsets using ergodic methods.

Regionally proximal relation of order d is an equivalence one for minimal systems and a combinatorial consequence