Let $a$ be an ultrafilter on a set $\mho$. Let $\mathcal{L}$ be an $a$-indexed family of filters (that is $\mathcal{L}$ is a function from $a$ to the set of filters on $\mho$).
Conjecture: For every nontrivial ultrafilter $a$ (2) does not follow from (1):
$\forall L \in \mho^a : ( ( \forall i \in a : L_i \in \mathcal{L}_i) \Rightarrow a \supseteq [L_i]_{i \in a})$ (where $[L_i]_{i \in a}$ is the filter generated by the family $L$ of sets);
$\forall L \in \mho^a : ( ( \forall i \in a : L_i \in \mathcal{L}_i) \Rightarrow \exists A \in a \forall i \in n : A \subseteq L_i)$.