In the following proof, (source https://jmp.sh/X5KwZ1Zt)
Using statement (i) as stated I only get
$X_{F^t}+D\varphi_F^tX_{G^t}\circ (\varphi^t_F)^{-1}=X_{F^t}+X_{{G^t}\circ (\varphi^t_F)^{-1}}$ but why is this equal to $ X_{F^t +G^t\circ (\varphi^t_F)^{-1}}$ ?
Edit:
Trying to prove that the map $M:F\mapsto X_F$ is linear:
$M(F)=X_F, M(G)=X_G, M(\alpha F+\beta G)=X_{\alpha F+\beta G}$
By definition of Hamiltonian vector field
$i_{X_F}\omega=dF$, $i_{X_G}\omega=dG$
and
$i_{X_{\alpha F +\beta G}}\omega=d(\alpha F +\beta G)=\alpha dF +\beta dG =\alpha i_{X_F}\omega +\beta i_{X_G}\omega$
But I don't know how to link this to $\alpha M(F) +\beta M (G)$