In our note we present a very simple and short proof of a new interesting fact about the faces of an integer hull of a given rational polyhedron. This fact has a complete analog in Linear Programming theory and can be useful to establish new constructive upper bounds on the number of vertices in integer hulls of $\Delta$-modular polyhedra, which can theoretically be useful for integer linear maximization problems with a convex or quasiconvex objective function. Additionally, we introduce the notion of \emph{deep bases} of a linear program. The problem to estimate their number by a non-trivial way seems to be quite challenging.