  A poset $\bfp$ is well-partially ordered (WPO) if all its linear extensions
are well orders~; the supremum of ordered types of these linear extensions is
the {\em length}, $\ell(\bfp)$ of $\bfp$. We prove that if the vertex set $X$
of $\bfp$ is infinite, of cardinality $\kappa$, and the ordering $\leq$ is the
intersection of finitely many partial orderings $\leq_i$ on $X$, $1\leq i\leq
n$,
  then, letting $\ell(X,\leq_i)=\kappa\multordby q_i+r_i$, with $r_i<\kappa$,
denote the euclidian division by $\kappa$ (seen as an initial ordinal) of the
length of the corresponding poset~:\[ \ell(\bfp)<
\kappa\multordby\bigotimes_{1\leq i\leq n}q_i+ \Big|\sum_{1\leq i\leq n}
r_i\Big|^+ \] where $|\sum r_i|^+$ denotes the least initial ordinal greater
than the ordinal $\sum r_i$. This inequality is optimal (for $n\geq 2$).
