A curious operation

From the Putnam Competition (I believe):

SupposeĀ # is a binary operation on a set S such that the following properties hold:

  1. For all a, b, c in S, (a#b)#c = a#(b#c);
  2. For all a, b in S, if a#b = b#a then a=b.

Prove that, for all x, y, z in S, we have x#y#z = x#z.

Solution

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