That assumes that every fraction has a unique simplest form. The first section of the article makes no such claims about the existence of a simplest form of fraction. The proof uses just algebraic manipulation, the fact that a sequence of strictly decreasing positive integers is finite in length, and the definition of a rational number (there exist integers p, q (q != 0) such that the number can be expressed as p/q).
t0mek sais "the simplest form", but the comment is fixable by changing that to "a simplest form". (For example, if hypothetically a/b and c/d where somehow the same number, and yet somehow there is no x such that a/b = xc/xd, the argument about how 2 divides into a and b also applies to c and d.)
