Of a special interest are tilings in hyperbolic n–space. In 1974, K. Böröczky published a construction of tilings of hyperbolic plane by a single prototile. It is natural to extend the study of tiling problems to the hyperbolic plane as well as hyperbolic spaces of higher dimension. This Böröczky construction can be extended to any dimension, yielding tilings of hyperbolic n–space. To obtain corresponding non face-to-face tiling of 3-space into convex "prismatic" equal hexa-faceted polyhedra it is enough every nine-faceted polyhedra of Böröczky’s tiling to cut into four prismatic polyhedra its "coordinate" planes of symmetry. The tilings (face-to-face and non-face-to-face) of n–dimensional hyperbolic space are under construction almost literally in the same way through partition of corresponding–horospheres into geodesic–cube (cubiliaj). An analogous construction works for arbitrary dimension. Theorem. In the hyperbolic n–space, there exists a non-regular non face-to-face (non-normal) tiling composed of congruent convex polyhedral tiles, which can’t be transformed into regular tiling using any permutation of the polyhedral tiles. The proposed construction can be considered and as the constructive proof of the theorem of the existence of non-face-to-face tilings in the n – dimensional hyperbolic space into equal, convex and compact polyhedra. The work outlined some possible generalizations of Böröczky’s construction, which in most cases, also allow to construct and non-face-to-face tilings. Features of tilings can constructively prove some general statements concerning, for example, point Delone Sets and Delone tilings. In the article it is also discussed the question of the number of hyperfaces for hyperbolic n– dimensional tile.