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If you looked at a lot of literature of the transgendered community (and by the way, they should probably be termed transgendered and not transsexual), you will find the idea of an essential nature of being a man or a woman is not common. Read the works of Kate Bornstein, for an example. Or essays by Sandy Stone. Or any other countless writers and theorists from a transgendered perspective. Indeed, it is the not the transgendered people who so often make arguments of being trapped in the wrong body, but the medical establishment that regulates who get the surgery and who doesn't. In order to get the surgery, a trans person is often expected to recite a certain a narrative, to explain certain feelings. Often these have little to nothing to do with what the trans person feels or has experienced. Sandy Stone's classic essay "The Empire Strikes Back: A PostTranssexual Manifesto" (which can be here http://www.actlab.utexas.edu/~sandy/empire-strikes-back ) details some of this.
On the other hand, the Mary Dalys and Janice Raymonds of the world seem completely fine with gender essentialism. Their problems with trans people seem even to be how it destabilizes our notions of gender.