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It is given an alternative self-contained proof of the homogenization theorem for periodic multiparameter integrals that was established by the authors in  . The proof in that paper relies on the so-called compactness method for Γ-convergence while the one presented here is by direct verification: the candidate to be the limit homogenized functional is first exhibited, the definition of Γ-convergence is then verified. This is done by extension of bounded gradient sequences using the Acerbidoi:10.1017/s0308210500003498 fatcat:luhoocmp3zbm7nwqtyhbsrge6a