The elementary approach to the theory of area, described in section 4.1, includes definitions of polygonal region, of equivalence of a pair V, W of regions under decomposition into the same finite number n of respectively congruent subregions with disjoint interiors, and of the area of a polygonal region. It includes the Bolyai–Gerwien theorem: polygonal regions are equivalent if and only if they have the same area. The 1924 papers of Alfred Tarski and Stefan Banach, translated in chapters 5 and 6, extended that theory to include analogous results with a different, set-theoretic, definition of equivalence.