Zero Vector Space Product iff Factor is Zero/Proof 2

Theorem
Let $\left({G, \circ}\right)$ be a group whose identity is $e$.

Let $\left({K, +, \circ}\right)$ be a division ring whose zero is $0$ and whose unity is $1$.

Let $\left({G, +_G, \circ}\right)_K$ be a $K$-vector space.

Let $x \in G, \lambda \in K$.

Then $\lambda \circ x = e \iff \left({\lambda = 0 \lor x = e}\right)$.

Proof
The sufficient condition is proved in Vector Scaled by Zero is Zero Vector, and in Zero Vector Scaled is Zero Vector.

The necessary condition is proved in Vector Product is Zero Only If Factor is Zero.