We show that critical percolation on a product of two regular trees of degree
$\ge$ 3 satisfies the triangle condition. The proof does not examine the
degrees of vertices and is not "perturbative" in any sense. It relies on an
unpublished lemma of Oded Schramm.Comment: Published in at http://dx.doi.org/10.1214/10-AOP618 the Annals of
  Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
  Statistics (http://www.imstat.org