  A new scheme for proving pseudoidentities from a given set {\Sigma} of
pseudoidentities, which is clearly sound, is also shown to be complete in many
instances, such as when {\Sigma} 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 {\Sigma} defines a pseudovariety V which is {\sigma}-reducible for
the equation x=y, provided {\Sigma} is enough to prove a basis of identities
for the variety of {\sigma}-algebras generated by V. This gives ample evidence
in support of the conjecture that the proof scheme is complete in general.
