  We investigate a model theoretic invariant $\kappa_{srd}^m(T)$, which was
introduced by Shelah in his famous book, and prove that $\kappa_{srd}^m(T)$ is
sub-additive.
  When $\kappa_{srd}^m(T)$ is infinite, this gives the equality
$\kappa^m_{srd}(T)=\kappa^1_{srd}(T)$, answering a question by Shelah. We apply
the same proof method to analyze another invariant $\kappa^m_{ird}(T)$, and
show that it is also sub-additive, improving a result in the book.
