The motivation to this question can be found in:

About equivalent statements of the Birch and Swinnerton-Dyer Conjecture

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? Is this true for curves over rationals?