Evaluation Linear Transformation is Bilinear

Theorem
Let $R$ be a commutative ring.

Let $G$ be an $R$-module.

Let $G^*$ be the algebraic dual of $G$.

Let $\innerprod x t$ be the evaluation linear transformation from $G$ to $G^{**}$.

Then the mapping $\phi: G \times G^* \to R$ defined as:
 * $\forall \tuple {x, t} \in G \times G^*: \map \phi {x, t} = \innerprod x t$

satisfies the following bilinearity properties:

Proof
and: