as is well known, we can put a metric on the upper half plane $\mathbb{R}^+ \times \mathbb{R}$ by setting $$ d\left((x,t);(x',t')\right):=\log\left(\frac{1 + \delta}{1 - \delta}\right)^{1/2}, $$ where $$ \delta := \left(\frac{(x'-x)^2 + (t'-t)^2}{(x'-x)^2+(t'+t)^2}\right)^{1/2}. $$

My question is: what is the most elementary proof that this is indeed a metric, meaning that the triangle inequality is satisfied?

Elementary means that no geometric arguments can be used (e.g. from hyperbolic geometry).

Thanks for your answers!

Edit: I am looking for computational solutions which do not make use of any invariance property.

notwant to make use of the invariance property? $\endgroup$4more comments