well-defined map: transvections in symmetric space

Hi!

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 $M$ be a symmetric space with involutions $s_p$, transvections $\tau_v$ and a point $p_0 \in M$. Then: $\tau_p(q)$ depends smoothly on $(p, q) \in M \times M$ where $\tau_p$ is $\tau_v$ such that $\tau_v(p_0) = p.$
(Let me give you the definition for $\tau_v$: Let M be a symmetric space and $c:\mathbb R \rightarrow M$ a geodesic with $p = c(0), v = \dot c(0).$ Then for $t \in \mathbb R$ the isometries

$\tau_{tv} = s_{c(t/2)}\circ s_{c(0)}$

are called transvections.)

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