Proposition 2

*Let $X$ be a manifold of class $C^r$, $m$ a law of composition of class $C^r$ on $X$ and $e$ an identity element for $m$.

(i) *The vector $0_e$ is an identity element for $T(m)$. (ii) *$T_e(X)$ is stable under $T(m)$ and the law of composition induced on $T_e(X)$ by $T(m)$ is the vector space addition on $T_e(X)$. (iii) *Let $U$ be an open subset of $X$ and $\alpha$ a mapping of class $C^r$ of $U$ into $X$ such that, for all $x \in U$, $\alpha(x)$ is the inverse of $x$ under $m$. Then, for all $u \in T(U)$, $T(\alpha)u$ is the inverse of $u$ under $T(m)$.

Properties (3) and (4) show that $T(m)(0_e, u) = T(m)(u, 0_e) = u$ for all $u \in T(X)$, whence (i). For $u, v$ in $T_e(X)$,

whence (ii). Finally the relations $m(x, \alpha(x)) = m(\alpha(x), x) = e$ for all $x \in U$ imply

for all $u \in T(U)$, whence (iii).