  We prove a uniqueness theorem for a large class of functional equations in
the plane, which resembles in form a classical result of Aczel. It is also
shown that functional equations in this class are overdetermined in the sense
of Paneah. This means that the solutions, if they exist, are determined by the
corresponding relation being fulfilled not in the original domain of validity,
but only at the points of a subset of the boundary of the domain of validity.
