The Dark Knight--Batman vs. Joker question (SPOILERS)?

I have to write a paper on The Dark Knight, and I need a few questions answered. So basically, the whole movie, Joker is "watching the world burn". But his purpose--what he really wants--is to prove that no one is "good" or "incorruptible". Right? So that's why Batman is so fun for Joker to fight, because he... show more I have to write a paper on The Dark Knight, and I need a few questions answered. So basically, the whole movie, Joker is "watching the world burn". But his purpose--what he really wants--is to prove that no one is "good" or "incorruptible". Right? So that's why Batman is so fun for Joker to fight, because he doesn't go down easily. But Batman has this "one rule" and that is, he won't kill anyone. He didn't kill the Joker when he had those 2 chances to do so, and didn't kill anybody else. So the Joker "wins" the fight if Batman breaks down and compromises on his rule, proving that he is corrupt, and Batman "wins" if he can stop Joker by turning him in to the law (which would probably result in the death penalty if they could ever keep him in jail that long) and still keep his morals intact. But at the end of the movie, it seems that Batman does compromise by killing Two Face in order to save Commissioner Gordon's son. So does that mean that Joker "won" in the end? Is Two-Face really dead? I just really want to know if Batman was finally corrupted. Because then Joker would have proved his point there. Is there a bigger picture than this? I mean, is there someway to excuse the killing of Two-Face? And on a side note, why did Joker want Batman to remove the mask? He didn't want to kill him, and he was already targeting loved ones...? Just curiosity? Feel free to correct anything I've said here, because it's just speculation. Thanks!
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