In the paper "On discrete subgroups of the two by two projective linear group over p-adic fields"
https://projecteuclid.org/download/pdf_1/euclid.jmsj/1260541107
in section 1 when presenting the axiom (G, l, I), he says that the axiom he gives is sufficient to show that we can define the double coset ring with respect to $U$ and $G$ ($U$ being a subgroup of $G$), which I would assume is equivalent to saying that the product of any two double cosets of $U$ is a finite union of double cosets.
I'm having a bit of difficulty in seeing how this follows from the axiom given, would welcome any tips.
