Free will is a topic as old as the hills. We all act as though we’ve got it, but established science gives this slim probability.

So does the question of free will matter? Certainly. Just not when it comes to morals and justice. Systems of morals and justice exist to guide behaviour in beneficial ways, and to the extent that they succeed at this, free will is irrelevant. <..> Our relationship to anti-social behaviour may be much more effective, and more compassionate, if we gave up the extremely naive notion of free choice that we currently assume — a notion which has already been made obsolete by experimental evidence of just how much of our behaviour is directed by unconscious, autonomous parts of ourselves.

Apparently, free will has been proven more than likely not to exist. How correct is this assertion? What proof is there to it from, quote, "established science"?