Proof: All of these results are special cases of the cancellation law for an associative operation (theorem 2.19).

Proof: These are special cases of the remark made earlier that an identity element is always invertible, and is its own inverse.

Proof: These are special cases of theorem 2.17.

I will now start the practice of calling a field . If I say `` let be a
field" I assume that the operations are denoted by and .

Proof: We know that , and hence

Also, , so

By the cancellation law for addition, .

Proof:

**Case 1:**- Suppose . Then (2.69) is true because every statement implies a true statement.
**Case 2:**- Suppose . By theorem 2.66, , so

Since , we can use the cancellation law for multiplication to get

and hence

Thus (2.69) holds in all cases.

Proof: Let be elements in . Then since multiplication is
commutative, we have

By the distributive law,

Since is the multiplicative identity,

and hence

By the cancellation law for addition

By commutativity of multiplication and the distributive law,

and

Since is the multiplicative identity and addition is associative

and hence

Since multiplication is commutative

and by the cancellation law for addition,

Hence, is commutative.

Proof:

Proof: Let . By (2.74) it is sufficient to prove

Well,

I'll call the set

the set of

and

Here should not be confused with .

In general, if , I define

Then for all digits

so, for example

and you can see that .

Proof:

Also,