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Professor X controls minds.

Neo controls the Matrix, a mind-control device.

Inside the Matrix whose power is greater? Obviously Xavier's first order of business would be giving himself working legs, but after that who wins? They face off in an empty city.

Professor X easly.

I don't quite get the circumstances, are they fighting in a way that allows Xavier (thanks to his telepathy) to in some way control the matrix just like Neo can, or is it a pure Telepathic struggle between the two?

I don't think Xavier would be able to patch into the Astral Plane so easily in the Matrix if at all.