Some partitions associated with a partially ordered set

For any partially ordered set P, let $\text{d}{}_{\mathrm{k}}\text{(P)(}\text{d}\text{?}{}_{\mathrm{k}}\text{(P))}$ denote the cardinality of the largest subset of P obtained by taking the union of k antichains (chains). Then there exists a partition Δ = {Δ_l ? Δ₂ > … ? Δ_l} of | P | such that d_k(P) = Δ₁ + Δ₂ + … + Δ_k and $\text{d}\text{?}{}_{\mathrm{k}}\text{(P) = Δ}{}_1{}^1\text{+ Δ}{}_2{}^1\text{+ … + Δ}{}_{\mathrm{k}}{}^1$ for each k, where $\text{Δ}{}^1$ denotes the partition conjugate to Δ. This result can be used to prove a general class of “Dilworth-type” theorems for subfamilies of P.