Vincent Law is Ergo Proxy, who is Proxy 1. Proxy one divided himself into two. That which is divided must become one. So Proxy 1 splits into 2, and one half leaves the dome, the other half remains. The half that leaves gets rid of their memory... who are they now? They are no longer the true Proxy Project creation. They are now gaining self-awareness and forging a new identity.
It is perhaps because they rejected their imperfect origin that they decided to dump their memory, and become Vincent Law. Victory over Law. He has claimed victory over the rules. He is no longer just a proxy. The amnesia effect allows us to follow him on his journey against the strongest enemy he'll ever face... himself. A journey we all must take if we strike out against what we were planned to be and instead seek out our own destiny.
So Proxy 1 is Ergo Proxy... He is both one and the same, and he must embrace every aspect of himself if he is to truly choose his own future. It's not an easy thing to do, but he has Re-l, or reality. He also has pino, and "in vino, veritas", or "in wine there is truth".
So moving forward with reality and truth, the lab project known as Proxy One will eventually embrace that he is Ergo Proxy, and yet overcome his origins and finally reach happiness by redefining himself as Vincent Law, victorious over the laws his very cells were programmed to follow. He felt the pulse of the awakening, and instead of dying, he awoke. He is the agent of death... or was... not anymore... He is Vincent Law, and Vince is in full control as he sails off with truth and reality, and a new autorave who perhaps represents the discipline he has finally achieved.