  We give an alternate proof of one of the inequalities proved recently for
martingales (=sums of martingale differences) in a non-commutative $L_p$-space,
with $1<p<\infty$, by Q. Xu and the author. This new approach is restricted to
$p$ an even integer, but it yields a constant which is $O(p)$ when $p\to
\infty$ and it applies to a much more general kind of sums which we call
$p$-orthogonal.
