One-Parameter Groups in a Group

In the setting of Routine Facts, Continued: Typed Arithmetic, Maps and Predicativity, let $(G,\cdot)$ be a group with identity $e$, unique by Uniqueness of the Identity Element and of Inverses in a Group (i), products written by juxtaposition as in Semigroups (i).

(i) One-parameter groups. A one-parameter group in $G$ is a function $T$ from $\mathbb{R}$ to $G$ such that $T(0)=e$ and

$$T(s+t)=T(s)\,T(t)\qquad\text{for all } s,t\in\mathbb{R}.$$