Addition Rule for Gaussian Binomial Coefficients/Formulation 1

Theorem
Let $\dbinom n m_q$ denote a Gaussian binomial coefficient.

Then:
 * $\dbinom n m_q = \dbinom {n - 1} m_q + \dbinom {n - 1} {m - 1}_q q^{n - m}$

Proof
By definition of Gaussian binomial coefficient:

Thus: