**Note**. We will keep all the notation and hypothesis given in this Definition.

**Lemma 1. **If is a left -module and then as -vector spaces.

*Proof*. Clearly both and are -vector spaces. Define and by and for all and Since is -balanced, it induces an abelian group homomorphism defined by for all and Obviously is also a -linear and and are identity maps. So is a -linear isomorphism.

**Lemma 2**. Let and suppose that are the set of left cosets of in Then as right -modules.

*Proof*. By definition, every element of is uniquely written as a finite -linear combination of elements of and every element of is in for some So To show that this sum is direct, suppose that

for some We need to prove that for all To do that, let and where and Then

Now if in (2), for some then and hence because the cosets are disjoint. So i.e. Thus, in (2), the elements are pairwise distinct and hence, since every element of is uniquely written as a -linear combination of elements of we get for all Therefore for all

**Theorem**. If is a representation of then

*Proof*. Let and suppose that are the set of left cosets of in Recall that tensor product distributes over direct sum. Thus by Lemma 2

and hence by Lemma 1