Size of Linearly Independent Subset is at Most Size of Finite Generator

Theorem
Let $R$ be a division ring.

Let $V$ be an $R$-vector space.

Let $F \subseteq V$ be a finite generator of $V$ over $R$.

Let $L \subseteq V$ be linearly independent over $R$.

Then $\left\vert{L}\right\vert \le \left\vert{F}\right\vert$.