I have a question about a property of Busemann functions on Hadamard spaces.
Let $X$ be a complete CAT($0$) space. If $r:[0, \infty) \to X$ is a geodesic ray, and $x\in X$ the Busemann function is defined to be $$ \beta_r (x) = \lim_{t\to \infty} d(r(t), x) - t$$
Now let $\xi \in \partial X$ and let $r$ and $\sigma$ be two geodesic rays starting at $x$ and $y$ respectively with $r(\infty) = \xi = \sigma(\infty)$.
The property I try to prove is the following: for any $z \in X$
$$ \beta_r(z) = \beta_\sigma(z)- \beta_\sigma(x)$$
I have been trying for some time now, but I couldn't prove it. Any help would be greatly appreciated!