a *right* invariant vector field —see Appendix A for the notations. In the language of , the differential system is $$\dot g(t) = \sum_{a=1}^r Z_a(t)\,{\gamma}^a_G(g(t)).
\label{eq:fund}$$ The theorem says that for every choice of the functions $Z_a(t)$ we have a (right invariant) Lie–Scheffers system on the Lie group $G$, and any such system is of this form. The reader should be aware, nevertheless, that for a system of the type  there might be more than one superposition rule [@TresMosqueterosbis]. The reason one needs only one particular solution is precisely the right invariance of the last equation: if $g(t)$ is the solution such that $g(t_0)=1$, then consider $\bar g(t):=g(t)\,g_0$ for each $g_0\in G$. We have $$\begin{aligned}
&&\dot{\bar g}(t) = T_{g(t)}R_{g_0}(\dot g(t)) =
T_{g(t)}R_{g_0}\left[T_1R_{g(t)}\left(\sum_{a=1}^r
Z_a(t){\gamma}^a\right)\right]
\cr
&&= T_1R_{g(t)g_0}\left(\sum_{a=1}^r Z_a(t){\gamma}^a\right) = \sum_{a=1}^r
Z_a(t){\gamma}^a_G(\bar g(t)).\end{aligned}$$ Equation $X^R(g) = T_1R_gX^R(1_G)$ has been used. Therefore $\bar
g(t)$ is the solution of the same equation  with $\bar
g(t_0)=g_0$: the solution curves of the system are obtained from just one of them by right-translations. In other words, the superposition functional  can be symbolically expressed by $\Psi(g_{(1)},k)=g_{(1)}k$, with $k\in G$; for which always $m=1$.

Lie–Scheffers systems live on manifolds which are not groups in general; however, they are always 