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A new scheme for proving pseudoidentities from a given set Σ of pseudoidentities, which is clearly sound, is also shown to be complete in many instances, such as when Σ defines a locally finite variety, a pseudovariety of groups, more generally, of completely simple semigroups, or of commutative monoids. Many further examples when the scheme is complete are given when Σ defines a pseudovariety V which is σ-reducible for the equation x=y, provided Σ is enough to prove a basis of identities fordoi:10.4171/pm/2012 arXiv:1708.09681v1 fatcat:pzw6pvxqhjhhtfymz4x6roxh7i