every uniformly continuous semigroup of operators can be extended to a uniformly continuous group

Question

I'm reading Vrabie and I'm in trouble with this remark.

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Given a uniformly continuous semigroup $\{S(t)\mid t\geq 0\}$, since $S(t)$ is invertible and can be extended to a uniformly continuous group via $G:X\longrightarrow X$ given by\begin{equation*} G(t)= \begin{cases} \left[S(-t)\right]^{-1} & \text{if}\; t<0 \\ S(t) & \text{if}\; t\geq 0 \end{cases} \end{equation*}.

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To see that is a uniformly continuous group we have to check:

1. $G(0)=I$
2. $\displaystyle\lim_{t\to 0} G(t)=I$
3. $G(s+t)=G(s)G(t)$

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1. $G(0)=S(0)=I$
2. $\displaystyle\lim_{t\to 0^-} G(t)=\displaystyle\lim_{t\to 0^-} [S(-t)]^{-1}=I=S(0)=\displaystyle\lim_{t\to 0^+} G(t)$ since $S(t)$ is invertible .
3. If $s,t\geq 0$ then $$G(s+t)=S(s+t)=S(s)S(t)=G(s)G(t)$$ If If $s,t\leq 0$ then $$G(s+t)=[S(-s+(-t))]^{-1}=[S(-s)S(-t)]^{-1}=S(-t)^{-1}S(-s)^{-1}=G(t)G(s)$$ If $t\geq 0$ and $s<0$ with $t+s\geq 0$ $$G(s+t)=S(s+t)$$ but since $t<0$ I can't write $S(s+t)=S(s)S(t)$. And I can't prove 3.