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given this block diagram:

enter image description here

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:

  1. bottom path: take input at $n$, multiply by $-1$ and delay by $1$, gives: $-1x[n-1]$
  2. top path: take input at $n$ and sum it by result of bottom path, gives: $x[n] + -1x[n-1]$
  3. 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?

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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

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

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