Prove that
length(P x Q) = length(P) + length(Q)
where P, Q are ordered sets of finite length.
Any ideas of where to start...
If and are chains in P and Q, respectively, then is a chain in P x Q, so length(P x Q) >= length(P) + length(Q). Conversely, if you have a chain in P x Q, then for each you have or (or both). So, you can construct chains in P and Q whose total length is >= n. This means that length(P) + length(Q) >= length(P x Q).