The motivation to this question can be found in:

http://mathoverflow.net/questions/125050/about-equivalent-statements-of-the-birch-and-swinnerton-dyer-conjecture?rq=1

My question is about the last equivalences:

$\mathrm{ord}_{s=1} L(E/K,s) = \mathrm{rank} (E/K) \iff |Ш| < \infty \iff 
|Ш_l^{\infty}| < \infty$ for some $l \iff \mathrm{ord}_{s=1} L(E/K,s) \leq \mathrm{rank} (E/K)$

How the equality in the first case is equivalent to the inequality in the last case?