Let G be a simply connected nilpotent Lie group and assume {mu(t)((i))}(t greater than or equal to 0) (i = 1, 2) are Poisson semigroups of probability measures on G with boundedly sup(1) (2)1 then P(1) (2) for all t > 0. ported Levy measures. We prove that if mu(1)((1)) = mu(1)((2)), then mu(t)((1)) = mu(t)((2)) for all t greater than or equal to 0. As a consequence, e.g. a convergent triangular system of rowwise i.i.d. probability measures on G which are concentrated on a fixed circular annulus automatically converges functionally.