Test Sets for Context Free Languages and Algebraic Systems of Equations over a Free Monoid

It is shown that for every context free language L there effectively exists a test set F, that is, a finite subset F of L such that, for any pair (g, h) of morphisms, g(x) = h(x) for all x in F implies g(x) = h(x) for all x in L. This result was claimed earlier but a detailed correct proof is given here. Together with very recent results on systems of equations over a free monoid this result implies that every algebraic system of equations is equivalent to a finite subsystem.