  Tucker and Ky Fan's lemma are combinatorial analogs of the Borsuk-Ulam
theorem (BUT). In 1996, Yu. A. Shashkin proved a version of Fan's lemma, which
is a combinatorial analog of the odd mapping theorem (OMT). We consider
generalizations of these lemmas for BUT-manifolds, i.e. for manifolds that
satisfy BUT. Proofs rely on a generalization of the OMT and on a lemma about
the doubling of manifolds with boundaries that are BUT-manifolds.
