Maximal Ideal WRT Filter Complement is Prime in Distributive Lattice/Lemma 4

Lemma for Maximal Ideal WRT Filter Complement is Prime in Distributive Lattice
Let $N = \set {x \in L: \exists m \in M: x \le m \vee a}$.

Then we have that:
 * $N \cap F = \O$

Proof
$x \in N \cap F$.

Then:
 * $x \in N$

so for some $m \in M$:
 * $x \le m \vee a$

Furthermore, $x \in F$.

So by the definition of a filter:
 * $m \vee a \in F$

But this contradicts our assumption that $\forall m \in M: m \vee a \notin F$.