We show that for each we have . Since , we have , and hence as is a right congruence holds. Now implies and we are done.

q.e.d.

1.11

The following lemma is necessary to prove Theorem 6

Since and is a right congruence, implies , and as we can conclude .

On the other hand we show that implies by showing by induction on

q.e.d.

1.11

**Proof of Theorem 6**:
1.11.1