To be more specific, I'm asking about a detail in the proof of this theorem (6.2.19 on Marker's Introduction to Model Theory). The proof is very straightforward, by induction: we show that if $a_1,\dots,a_k$ are independent over $A$, then $\operatorname{RM}(\bar{a}/A)=k$. In the inductive step, they show first that $\operatorname{RM}(\bar{a}/A) \geq k$, which I can understand. However, for the other inequality, they take again a minimal formula $\phi(v)$ witnessing the type's rank, and the following argument is carried out:
"Suppose that $\psi(v)$ is a formula such that $\psi(M) \subset \phi(M)$ and $\neg\psi(a)$, then it suffices to show that $\operatorname{RM}(\psi)<k$."
Why does it suffice? I would use that $\phi = (\phi \land \psi) \lor (\phi \land \neg \psi)$ to get some kind of contradiction if $\operatorname{RM}(\psi)\geq k $, but this doesn't seem to bound anything from above. My intuition is that, if removing the tuple $\bar{a}$ from $\phi(M)$, makes the rank drop then the rank originally could not be higher than $k$, but it is not clear to me just from the statement made in the book. Any help would be appreciated. The rest of the proof is very much understandable.
If the rank of $\phi$ is $(k+1)$ or greater, then this is witnessed by infinitely many disjoint definable subsets of rank $k$ or greater. But every definable subset of rank $k$ or greater contains $a$. So there are not even two disjoint definable subsets of rank $k$ or greater.