Now take two non-empty sets and of distinct non-consecutive Fibonacci numbers which have the same sum, . Consider sets and which are equal to and from which the common elements have been removed (i. e. and ). Since and had equal sum, and we have removed exactly the elements from from both sets, and must have the same sum as well, . Now we will show by contradiction that at least one of and is empty. Assume the contrary, i. e. that and are both non-empty and let the largest member of be and the largest member of be . Because and contain no common elements, . Without loss of generality, suppose . Then by the lemma, , and, by the fact that , , whereas clearly . This contradicts the fact that and have the same sum, and we can conclude that either or must be empty. Now assume (again without loss of generality) that is empty. Then has sum 0, and so must . But since can only contain positive integers, it must be empty too. To conclude: which implies , proving that each Zeckendorf representation is unique. Fibonacci multiplication One can define the following operation on natural numbers , : given the Zeckendorf representations and we define the Fibonacci product For example, the Zeckendorf representation of 2 is , and the Zeckendorf representation of 4 is ( is disallowed from representations), so (The product is not always in Zeckendorf form. For example, ) A simple rearrangement of sums shows that this is a commutative operation; however, Donald Knuth proved the surprising fact that this operation is also associative. Representation with negafibonacci numbers The Fibonacci sequence can be extended to negative index using the rearranged recurrence relation which yields the sequence of "negafibonacci" numbers satisfying Any integer can be uniquely represented as a sum of negafibonacci numbers in which no two consecutive negafibonacci numbers are used. For example: 0 is represented by the empty sum. , for example, so the uniqueness of the representation does depend on the condition that no two consecutive negafibonacci numbers are used. This gives a system of coding integers, similar to the representation of Zeckendorf's theorem. In the string representing the integer , the th digit is 1 if appears in the sum that represents ; that digit is 0 otherwise. For example, 24 may be represented by the string 100101001, which has