Multiplying "ab" = "c" on the left by "a" gives "b" = "ac". Multiplying on the right by "c" gives "bc" = "a". Multiplying "ab" = "c" on the right by "b" gives "a" = "cb". Multiplying "bc" = "a" on the left by "b" gives "c" = "ba", and multiplying that on the right by "a" gives "ca" = "b". After filling these products into the table, we find that the "ad" and "af" are still unaccounted for in the "a" row; as we know that each element of the group must appear in each row exactly once, and that only "d" and "f" are unaccounted for, we know that "ad" must equal "d" or "f"; but it cannot equal "d", because if it did, that would imply that "a" equaled "e", when we know them to be distinct. Thus we have "ad" = "f" and "af" = "d".