  In [J. Combin. Theory Ser. B 70 (1997), 2-44] we gave a simplified proof of
the Four-Color Theorem. The proof is computer-assisted in the sense that for
two lemmas in the article we did not give proofs, and instead asserted that we
have verified those statements using a computer. Here we give additional
details for one of those lemmas, and we include the original computer programs
and data as "ancillary files" accompanying this submission.
