converting block diagrams with delays to equations given this block diagram:



can someone please clarify how to interpret the second delay (the one prior to $Y$)? to convert this block diagram into a simple equation, I interpret the diagram as:


bottom path: take input at $n$, multiply by $-1$ and delay by $1$, gives: $-1x[n-1]$
top path: take input at $n$ and sum it by result of bottom path, gives: $x[n] + -1x[n-1]$
take sum of bottom and top path and delay it (?) - this is the part I'm confused about. it doesn't make sense to write: $(x[n] + -1x[n-1])[n-1]$ which is what the diagram looks like to me.


what is the right way to read this diagram?
If you are confused about the interpretation, why not divide the diagram into 2 parts. The signal to the left of the rightmost delay element is assumed to be z[n].  

Thus, now $$y[n] = z[n-1]$$  

$$z[n] = x[n] + (-1)\cdot x[n-1]$$  

Thus

$$y[n] = x[n-1]  -x[n-2]$$