On the theory of mainstay parallelohedra

In 1998 the first author announced a theorem stating that every primitive $n$-dimensional parallelohedron can be represented, up to an affine transformation, as a weighted Minkowski sum of parallelohedra belonging to a certain finite set of $n'$-dimensional $(n'\leqslant n)$ mainstay parallelohedra situated in a special way. This paper contains a detailed proof of this theorem in a refined and definitive form.