Crossed products by Hilbert pro-C*-bimodules versus tensor products

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We show that if (X.A) and (Y,B) are two isomorphic Hilbert pro-C -bimodules, then the crossed product A×ₓZ of A by X and the crossed product B×YZ of B by Y are isomorphic as pro-C-algebras. We also prove a property of "associativity" between " ⊗" and "×ₓ" as well as " ⊗" and "×ₓ". As an application of these results we show that the crossed product of a nuclear pro-C -algebra A by a full Hilbert pro-C-bimodule X is a nuclear pro-C-algebra.