Projections of $k$-dimensional subsets of $\mathbf R^n$ onto $k$-dimensional planes

Some properties of projections of sets with non-vanishing Hausdorff $k$-measure onto $k$-planes are studied. It is stated that there is a wide class of $k$-planes in $\mathbf R^n$ such that a projection of a closed $k$-dimensional set onto any plane of that class has dimension equal to $k$.