  We give a simple crystal theoretic interpretation of the Lascoux's expansion
of a non-symmetric Cauchy kernel $\prod_{i+ j\leq n+1}(1-x_iy_j)^{-1}$, which
is given in terms of Demazure characters and atoms. We give a bijective proof
of the non-symmetric Cauchy identity using the crystal of Lakshmibai-Seshadri
paths, and extend it to the case of continuous crystals.
