Height of a superposition

We observe that, given a poset (E, R) and a finite covering R = R-1. boolean OR. . .boolean OR. R-n of its ordering, the height of the poset does not exceed the natural product of the heights of the corresponding sub-relations: h(E, R) <= h(E, R-1) circle times. . .circle times. h(E, R-n). Conversely for every finite sequence (xi(1), . . ., xi(n)) of ordinals, every poset (E, R) of height at most xi(1) circle times. . . circle times epsilon(n) admits a partition (R-1, . . ., R-n) of its ordering R such that each (E, R-k) has height at most xi(k). In particular for every finite sequence (xi(1), . . . , xi(n)) of ordinals, the ordinal xi(1) (circle times) under bar. . . (circle times) under bar := sup {(xi(1)' circle times. . .circle times xi(n)') + 1 : xi(1)',. . ., xi(n)' < xi(n)} is the least. for which the following partition relation holds h xi ->(h(xi 1), . . , h(xi n))(2) meaning: for every poset (A, R) of height at least. and every finite covering (R-1, . . ., R-n) of its ordering R, there is a k for which the relation (A, R-k) has height at least xi(k). The proof will rely on analogue properties of vertex coverings w.r.t. the natural sum.