Zero Vector Space Product iff Factor is Zero/Proof 1

Proof
A vector space is a module, so all results about modules also apply to vector spaces.

So from Scalar Product with Identity it follows directly that $\lambda = 0 \lor x = e \implies \lambda \circ x = e$.

Next, suppose $\lambda \circ x = e$ but $\lambda \ne 0$.

Then from Scalar Product with Identity: