= Solution
If $H|_L$ is constant, then $dH(v)=0$ for every $v\in TL$. By the definition of the <Hamiltonian vector field>,
$$
\omega(X_H,v)=-dH(v)=0.
$$
Since $L$ is <Lagrangian submanifold>[Lagrangian], $(TL)^\omega=TL$, so $X_H$ is tangent to $L$. Its <Hamiltonian isotopy>[Hamiltonian flow] preserves $L$. This is the <Hamiltonian flow preserves a constant-level Lagrangian> principle.
The cotangent lift is functorial:
$$
(f\circ g)_\#=f_\#\circ g_\#.
$$
Since $\phi_{t+s}=\phi_t\circ\phi_s$, it follows that
$$
\psi_{t+s}=(\phi_{t+s})_\#=(\phi_t)_\#\circ(\phi_s)_\#=\psi_t\circ\psi_s.
$$
Thus $(\psi_t)$ is a flow with infinitesimal vector field $V_\#$ as stated in the question.
Finally, a cotangent lift sends the <conormal bundle> $N^*Y$ to $N^*f(Y)$. Explicitly, if $\xi$ annihilates $T_xY$, then $(df_x^{-1})^*\xi$ annihilates $T_{f(x)}f(Y)$. Since $\phi_t(Y)=Y$, we have
$$
\psi_t(N^*Y)=N^*\phi_t(Y)=N^*Y,
$$
so the conormal bundle is invariant.
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