  We prove two lemmata about Schubert calculus on generalized flag manifolds
G/B, and in the case of the ordinary flag manifold GL_n/B we interpret them
combinatorially in terms of descents, and geometrically in terms of missing
subspaces. One of them gives a symmetry of Schubert calculus that we
christen_descent-cycling_. Computer experiment shows that these lemmata suffice
to determine all of GL_n Schubert calculus through n=5, and 99.97%+ at n=6. We
use them to give a quick proof of Monk's rule. The lemmata also hold in
equivariant (``double'') Schubert calculus for Kac-Moody groups G.
