the hyperbolic distance d(z1,z2)d(z1,z2) is the absolute value of the log of absolute value of the cross ratio between z1,z2z1,z2 and the two points of the h-line that goes through z1,z2z1,z2
Why is this so? Why is this the definition of hyperbolic distance? It looks like a formulae. But what is the "proof " of this?