# Property of greatest integer function

I came across the following mathematical statement in a proof. Can somebody tell me which property of greatest integer function makes it possible?

$x + y - \lfloor x + y \rfloor + z - \lfloor x + y - \lfloor x + y \rfloor + z \rfloor = x + y - \lfloor x + y \rfloor + z - \lfloor x + y + z \rfloor + \lfloor x + y \rfloor$

How do we get $\lfloor x + y - \lfloor x + y \rfloor + z \rfloor = \lfloor x + y + z \rfloor - \lfloor x + y \rfloor$?

In general, if $N$ is an integer, then $$\lfloor N+\alpha\rfloor=N+\lfloor\alpha\rfloor.$$
Proof : Let $N+\alpha=M+\beta$ where $M$ is an integer and $0\le \beta\lt 1$. Then, since we have $$\alpha=M-N+\beta\Rightarrow \lfloor\alpha\rfloor=M-N,$$ we have \begin{align}\lfloor N+\alpha\rfloor&=\lfloor M+\beta\rfloor\\&=M\\&=N+(M-N)\\&=N+\lfloor\alpha\rfloor.\end{align} Here, setting $N=-\lfloor x+y\rfloor,\alpha=x+y+z$ gives us $$\lfloor -\lfloor x+y\rfloor+(x+y+z)\rfloor=-\lfloor x+y\rfloor+\lfloor x+y+z\rfloor.$$