I'm trying to understand a proof for the fact that the isometry group of a symmetric space is a Lie group. The proof uses a lemma and I don't see how the lemma works. Here is the statement in question:

(Let me give you the definition for [itex]\tau_v[/itex]: Let M be a symmetric space and [itex]c:\mathbb R \rightarrow M[/itex] a geodesic with [itex]p = c(0), v = \dot c(0).[/itex] Then for [itex]t \in \mathbb R[/itex] the isometries

[itex]\tau_{tv} = s_{c(t/2)}\circ s_{c(0)}[/itex]

are called transvections.)

I don't see how the map [itex](p, q) \mapsto \tau_p(q)[/itex] is even well-defined. There can be more than one transvection mapping [itex]p_0[/itex] to [itex]p[/itex], and different transvections will in general give different values [itex]\tau_p(q)[/itex].