  The non-bijective version of Wigner's theorem states that a map which is
defined on the set of self-adjoint, rank-one projections (or pure states) of a
complex Hilbert space and which preserves the transition probability between
any two elements, is induced by a linear or antilinear isometry. We present a
completely new, elementary and very short proof of this famous theorem which is
very important in quantum mechanics. We do not assume bijectivity of the
mapping or separability of the underlying space like in many other proofs.
