This paper presents a decomposition theorem for partially ordered sets (posets) by R. P. Dilworth. The main theorem, Theorem 1.1, states that if every set of k+1 elements in a poset P is dependent, while at least one set of k elements is independent, then P can be expressed as the set sum of k disjoint chains. This theorem is a generalization of the Radó-Hall theorem on set representatives and has applications in distributive lattices. The paper also proves an imbedding theorem for finite distributive lattices, showing that such a lattice can be embedded into a direct union of k chains, where k is the maximum number of elements covering any given element. The proof begins with the case of finite posets, using induction on the maximal number of independent elements. It shows that if a poset can be decomposed into k disjoint chains, then adding a new element not in these chains can still maintain the decomposition. The proof involves analyzing the relationships between elements in chains and their comparability with a new element. It uses a contradiction argument to show that if a set of k elements is independent, then adding a new element would create a set of k+1 independent elements, violating the theorem's hypothesis. This leads to the conclusion that the maximal number of independent elements in a modified set is less than k, allowing the decomposition to continue. The paper also discusses a dual argument for the case of elements below the new element, showing similar results. The overall approach uses the properties of chains and independent sets to decompose the poset into disjoint chains, demonstrating the theorem's validity. The results have implications for understanding the structure of posets and their relationships to lattices.This paper presents a decomposition theorem for partially ordered sets (posets) by R. P. Dilworth. The main theorem, Theorem 1.1, states that if every set of k+1 elements in a poset P is dependent, while at least one set of k elements is independent, then P can be expressed as the set sum of k disjoint chains. This theorem is a generalization of the Radó-Hall theorem on set representatives and has applications in distributive lattices. The paper also proves an imbedding theorem for finite distributive lattices, showing that such a lattice can be embedded into a direct union of k chains, where k is the maximum number of elements covering any given element. The proof begins with the case of finite posets, using induction on the maximal number of independent elements. It shows that if a poset can be decomposed into k disjoint chains, then adding a new element not in these chains can still maintain the decomposition. The proof involves analyzing the relationships between elements in chains and their comparability with a new element. It uses a contradiction argument to show that if a set of k elements is independent, then adding a new element would create a set of k+1 independent elements, violating the theorem's hypothesis. This leads to the conclusion that the maximal number of independent elements in a modified set is less than k, allowing the decomposition to continue. The paper also discusses a dual argument for the case of elements below the new element, showing similar results. The overall approach uses the properties of chains and independent sets to decompose the poset into disjoint chains, demonstrating the theorem's validity. The results have implications for understanding the structure of posets and their relationships to lattices.