  By adding the total time derivatives of all the constraints to the Lagrangian
step by step, we achieve the further work of the Dirac conjecture left by
Dirac. Hitherto, the Dirac conjecture is proved completely. It is worth
noticing that the addition of the total time derivatives to the Lagrangian can
turn up some constraints hiding in the original Lagrangian. For a constrained
system, the extended Hamiltonian $H_E$ considers more constraints, and shows
symmetries more obviously than the total Hamiltonian $H_T$. In the Lagrangian
formalism, we reconsider the Cawley counterexample, and offer an example in
which in accordance with its original Lagrangian its extended Hamiltonian is
better than its total Hamiltonian.
