Cfrgtkky

I am trying to understand the proof of thm 37.16 of Jech on page 687.
I don't understand the first 4 lines, why does that suffice to proof the theorem? I don't see how they are related. It sais the following:

Assume MM and let ${A_i : i in W } $ be a maximal almost disjoint collection of stationary subsets of $omega_1$. We shall find a set $Zsubset W$ of size $leq aleph_1$ such that $sum_{iin Z} A_i$ contains a closed unbounded set. This implies the theorem.

Here the sum is the diagonal union. In the comments it was suggested that almost disjoint in this case means two different $A_i$'s have a non-stationary intersection.

Thanks!

Let's consider the Boolean algebra $Bbb B=mathcal P(omega_1)/rm NS$ (one can also just consider this as a notion of forcing). Saturation simply means that it has the $aleph_2$-chain condition.

So take a maximal family of mutually stationary sets, which is exactly a maximal antichain in $Bbb B$. If there is a family of size $aleph_1$ whose union contains a club, that means that this family would already constitute a partition of $1_{Bbb B}$ (in forcing terms: this family would induce a maxiaml antichain already), because $1_{Bbb B}$ is exactly the club filter on $omega_1$.

In other words, it means that every maximal antichain in $Bbb B$ is of size $<aleph_2$, which is exactly what we want to prove.