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The lemma is correct, and there are indeed no more than $n$ non-equivalent $f_{i}$. Here is a sketch of a proof: We can do a transformation such that the coefficient of (without loss of generality) $a_{1}^2$ is non zero, and by scaling, 1. Now by replacing $a_1$ with $a_1 - c_2 \cdot a_2 - \dotsb - c_n \cdot a_n$, where the coefficients $c_i$ are chosen appropriately, we can ensure that $a_{1}^2$ is the only place where $a_1$ appears, and now we can induct on the statement with $n-1$ variables.