Weak C-cleft extensions and weak Galois extensions

For a weak entwining structure (A,C,ψ) we formulate the notion of weak C-Galois extension with normal basis and we show that these Galois extensions are equivalent to the weak C-cleft extensions introduced in [J.N. Alonso Álvarez, J.M. Fernández Vilaboa, R. González Rodríguez, A.B. Rodríguez Raposo, Weak C-cleft extensions, weak entwining structures and weak Hopf algebras, J. Algebra 284 (2005) 679–704].