  The approach to the eta'-eta complex employing chirally well-behaved
quark-antiquark bound states and incorporating the non-Abelian axial anomaly of
QCD through the generalization of the Witten-Veneziano relation, is extended to
finite temperatures. Employing the chiral condensate has led to a sharp chiral
and U_A(1) symmetry restoration, but with the condensates of quarks with
realistic explicit chiral symmetry breaking, which exhibit a smooth, crossover
chiral symmetry restoration in qualitative agreement with lattice QCD results,
we get a crossover U_A(1) transition, with smooth and gradual melting of
anomalous mass contributions. This way we obtain a substantial drop of the eta'
mass around the chiral transition temperature, but no eta mass drop. This is
consistent with the present empirical evidence.
