I am looking at a problem in Guidry's Modern General Relativity, and the solution contains the following two sentences:
In the scalar product expression $A\cdot B = g_{\mu \nu}A^{\mu} B^{\nu}$, the left side is a scalar and $A$ and $B$ on the right side are vectors. Since the quantities, $g_{\mu \nu}$ contracted with tensors on the right side yield a tensor on the left side, by the quotient theorem $g_{\mu \nu}$ must define the components of a type (0,2) tensor.
For reference, the quotient rule says that if a set of quantities produces a tensor when contracted with a tensor, then that set of quantities is necessarily a tensor.
I don't understand this proof, specifically, why it says "the quantities, $g_{\mu \nu}$ contracted with tensors on the right side yield a tensor on the left side". If we apply tensors to both sides, wouldn't this only prove that $g_{\mu \nu}A^{\mu} B^{\nu}$ is a tensor, not $g_{\mu \nu}$?