I'm currently reading the "Primer on mapping class groups" by Farb-Magalit. Something that almost always comes up in the proofs is the "change of coordinates principle".
The first explanation of this principle is that for any non-separating simple closed curve $\alpha$, one can always find another non-separating simple closed curve $\beta$ such that $\alpha$ and $\beta$ fill our surface. That is, they are in minimal position and the cutsurface is a union of disks.
So far so good, but this fact is often used in other incarnations. For example for a sequence of isotopy classes of simple closed curves $a_1,...,a_n$ such that $i(a_i,a_j) = 0 \; \forall i \neq j$, we can find another sequence of isotopy classes of simple closed curves $b_1,...,b_{n-1}$ such that $i(a_i, b_i) = i(a_{i+1}, b_i) = 1$.
I just don't see how this follows from this first description of the change of coordinates principle.
In the book, they give the following incarnation of this principle, which looks like it has something to do with this construction on some kind of path of intersecting isotopy classes:
But I also don't fully understand what they want to tell me with this statement.
The question is the following:
How do you get from the first incarnation to this "path-like construction"? And what does this statement 6. tell us?