B. Associativity and Distributivity of Operations in

Thus for all ,

We will show that and are associative by using the usual properties of addition and multiplication on .

Proof:

- Case 1. Suppose . Then by our assumptions,

and

So

Since , it follows that and since is an integer , so . Then , so . - Case 2. If , use Case 1 with and interchanged.

Proof: Let
. Then

By adding to both sides of (B.5), we get

and by adding to both sides of (B.6), we get

Replace in (B.9) by its value from (B.7) to get

and replace in (B.10) by its value from (B.8) to get

By (B.11) and (B.12) and the associative law in ,

the associativity of follows from lemma (B.3).

Proof: The proof is nearly identical with the proof that is associative.

Proof: We have

Multiply both sides of (B.15) by to get

Replace in (B.19) by its value from (B.16) to get

Now add equations (B.17) and (B.18) to get

We know that for some ,

and if we substitute this into (B.21), we obtain

From (B.20) and (B.22) and the distributive law in , we conclude

The distributive law follows from lemma B.3.