Proposition 15

If E is a vector G-bundle of class C^r, τE is a vector G-bundle of class C^r.

Let g_0, x_0, U, M, N be as in the paragraph following Definition 4. Then the mapping (g, x) ↦ τ(ψ_{g,x}) of U into $\mathscr{L}(\tau M, \tau N)$ is the composition of the mapping (g, x) ↦ ψ_{g,x} of U into $\mathscr{L}(M, N)$ and the mapping f ↦ τ(f) of Isom(M, N) into Isom(τM, τN); these two mappings are of class C^r and hence so is their composition, whence the proposition.