An equivalence relation on

- (R) For any , we have , thus .
- (S) Let and be two elements of
. We have:

- (T) Let , and be three elements of
. We have:

By addition we have and by Proposition 1.15 we have , i.e. .

For example, and . The multipication can be also denoted without any symbol, like . Propositions 2.2, 2.3, 2.4 and 2.5 are very easy to prove; we leave the task to the reader.

- (i)
- .
- (ii)
- .

- (i)
- .
- (ii)
- .

- (i)
- .
- (ii)
- .

- (i)
- .
- (ii)
- .

By addition we have , i.e. and this equivalent to Because of the commutativity of the multiplication, we can show only the following property instead of the second equation above:

We leave the task to the reader.