Another proof of dilworth's decomposition theorem

Dilworth's famous theorem 1] states that if the maximal sized antichains of a finite poset X have n elements, then X can be covered by n chains. The number n is called the width of X. Apart from proofs relating the theorem to other key theorems of combinatorics (see 1–4]), there have been a number of direct proofs (see 1, 2, 5, 6]). The shortest of these is the one by Perles 5], the outline of which is as follows.