Proposition 11

Let $G_1$ and $G_2$ be Lie group germs and $X_1$ and $X_2$ manifolds of class $C^r$ ($r \geq 2$). For $i = 1, 2$, let $\psi_i$ be a law chunk of left operation of class $C^r$ of $G_i$ on $X_i$ and $D_i$ the associated law of infinitesimal operation. Let $\mu : G_1 \to G_2$ be a morphism and $\phi : X_1 \to X_2$ a mapping of class $C^r$. Suppose that, for all $a \in L(G)$, the vector fields $(D_1)_a$ and $(D_2)_{L(\mu)a}$ are $\phi$-related (Differentiable and Analytic Manifolds, R, 8.2.6). Then there exists a neighbourhood $\Omega$ of $\{e\} \times X_1$ in $G_1 \times X_1$ such that $\phi(\psi_1(g, x)) = \psi_2(\mu(g), \phi(x))$ for all $(g, x) \in \Omega$.

Let $p_1 : G_1 \times X_2 \to G_1, p_2 : G_1 \times X_2 \to X_2$ be the canonical projections. For all $(g_1, x_2) \in G_1 \times X_2$, let $f_{g_1, x_2}$ be the mapping $g_1 a \mapsto (D_2)_{L(\mu)a}(x_2)$ of $T_{g_1}(G_1) = g_1 L(G_1)$ into $T_{x_2}(X_2)$. The $f_{g_1, x_2}$ define a morphism of $p_1^*T(G_1)$ into $p_2^*T(X_2)$.

Let $x_0 \in X_1$. There exist an open neighbourhood $G$ of $e$ in $G_1$ and an open neighbourhood $X$ of $x_0$ in $X_1$ such that $\psi_1(g, x)$ and $\psi_2(\mu(g), \phi(x))$ are defined for $(g, x) \in G \times X$. We write, for $(g, x) \in G \times X$,

If $G$ and $X$ are sufficiently small, then, for all $(a, g, x) \in L(G_1) \times G \times X$,

Hence for $x \in X$, the morphisms $g \mapsto \alpha(g, x)$ and $g \mapsto \beta(g, x)$ are integrals of $f$; as

for all $x \in X$, it follows from Differentiable and Analytic Manifolds, R, 9.3.7, that $\alpha$ and $\beta$ coincide on a neighbourhood of $(e, x_0)$. Hence the proposition.