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By adding the total time derivatives of all the constraints to the Lagrangianstep by step, we achieve the further work of the Dirac conjecture left byDirac. Hitherto, the Dirac conjecture is proved completely. It is worthnoticing that the addition of the total time derivatives to the Lagrangian canturn up some constraints hiding in the original Lagrangian. For a constrainedsystem, the extended Hamiltonian $H_E$ considers more constraints, and showssymmetries more obviously than the total Hamiltonian $H_T$. In the Lagrangianformalism, we reconsider the Cawley counterexample, and offer an example inwhich in accordance with its original Lagrangian its extended Hamiltonian isbetter than its total Hamiltonian.

The Dirac Conjecture and the Non-uniqueness of Lagrangian