Consider the Banach Space $$ X=\{f:\mathbb{R} \longrightarrow \mathbb{R} ; f \:\mbox{is bounded and uniformly continuous \} }$$ with norm $$||f||= \sup_{x\in\mathbb{R}} |f(x)|, \: \forall\: f\in X.$$
For $ f\in X$ define $$(T(t)f)(s)=f(t+s), \: \forall \:t\geq 0 \: \mbox{and} \: s\in\mathbb{R}.$$
I want to show that $\{T (t) \; ; \; t\geq 0 \}$ is a $C _0$ semigroup, in X. I managed to show all the conditions of the definition of $C _0$ a semigroup, except that $$ T(t+\overline{s})f)(s)= T(t)f(s)\;T(\overline{s})f(s),\:\forall\:t,\overline{s}\geq0, f\in X, s\in\mathbb{R} ,$$ that is, $$ f(t+\overline{s}+s)=f(t+s)\;f(s+\overline{s}),\:\forall\:t,\overline{s}\geq0, f\in X, s\in\mathbb{R}.$$
How to show it?