Generalized Comparison between two projection in a von Neumann algebra

In Kadison-Ringrose vol II, the authors proved the Comparison Theorem (6.2.7). They describes it as a "generalization" of the corresponding theorem (6.2.6) that hold in a factor. But I cannot fully grasp the way they used this theorem in the subsequent sections. For example, they used the Comparison Theorem (6.2.7) in the following proposition in the book:

6.4.6(ii). Proposition Let $E, F$ are two projection in a von Neumann algebra $\mathscr{R}$ and $E$ is abelian such that $C_E \le C_F$. Then $E\precsim F$.

Proof. If $E\npreceq F$, then, from the comparison theorem there is some (non-zero) central subprojection $P$ of $C_E$ such that $PF\prec PE$...

I cannot understand the way they use the comparison theorem here in the first line of the proof. How exactly does the existence of the $P(\le C_E)$ follow from Comparison theorem?

What I understand is that: since $E\npreceq F$, neither $E\sim F$ nor $E\prec F$. So if we apply Comparison theorem 6.2.7 on these $E$ and $F$ we can say that (in the notaion of Theorem 6.2.7) $Q$ obtained from comparison is not equal to $I$...But I cannot see further .... Any explanation regarding the proof 6.4.6 (or Comparison Theorem) will be appreciated. Thanks.

The Comparison Theorem says that you can always find pairwise orthogonal central projections $P,Q,R$ such that $P+Q+R=I$ and $$ PF\prec PE,\qquad QF\sim QE,\qquad RE\prec RF. $$ When you have $E\not\preceq F$, you can conclude that $P\ne0$, which is what they use. This is because, if $P=0$, then $Q+R=I$ and you have that $E\preceq F$, a contradiction.