The aim of this paper is to derive simplest consequences of the author's result of the multidimensional invariance principle. We obtain bounds for the rate of strong Gaussian approximation of sums of independent R ^d-valued random variables ξ_j having finite moments of the form EH (||_j||), where H (x) is a monotone function growing not slower than x² and not faster than e^cx. A multidimensional version of results of Sakhanenko is obtained. Bibliography: 19 titles.