To see why a row or column cannot contain the same element more than once, let "a", "x", and "y" all be elements of a group, with "x" and "y" distinct. Then in the row representing the element "a", the column corresponding to "x" contains the product "ax", and similarly the column corresponding to "y" contains the product "ay". If these two products were equalthat is to say, row "a" contained the same element twice, our hypothesisthen "ax" would equal "ay". But because the cancellation law holds, we can conclude that if "ax" = "ay", then "x" = "y", a contradiction. Therefore, our hypothesis is incorrect, and a row cannot contain the same element twice. Exactly the same argument suffices to prove the column case, and so we conclude that each row and column contains no element more than once. Because the group is finite, the pigeonhole principle guarantees that each element of the group will be represented in each row and in each column exactly once.