I do not see a problem here, you just want to go from the right to the left and then from the left to the right. Only basic algebraic manipulation is required.

Proof

Let

Suppose . Multiplying both sides by we obtain:

Thus we have

For the converse, suppose . Subtracting from both sides, we obtain: . Now , since , thus we must have . But this means .

Thus we have .

Therefore, if , .

QED