Some doubt about minimal antichain cover of poset.

Question

Suppose $P$ is finite partially ordered set (poset) with $\preceq $. Suppose it's height is $n$ i.e the minimal number of antichains which cover $P$. Say $$\mathcal{A} = \{A_1,A_2,...A_n\}\;\;\;\;\;\; {\rm and}\;\;\;\;\;\;\mathcal{A}' = \{A'_1,A'_2,...A'_n\}$$ are two families of antichains which covers $P$. Suppose that $$|A_1|\leq |A_2|\leq ...\leq |A_n|$$ and $$|A'_1|\leq |A'_2|\leq ...\leq |A'_n|$$ Can we say that $|A_i|=|A'_i|$ for each $i\leq n$?

Answer 1

No, take $P = \{ a, a', b, b', x, y \}$ with order $a < a'$, $b < b'$.
Height is 2. P is covered by { a, b }, { a', x, y, b' }
and { a', x, b }, { a, y, b' }.