Basis of Vector Space Injects into Generator
Let $K$ be a division ring.
Let $V$ be a vector space over $K$.
Let $B$ be a basis of $V$.
Let $G$ be a generator of $V$.
Then there exists an injection from $B$ to $G$.
By Vector Space has Basis between Linearly Independent Set and Spanning Set, there exists a basis $C \subset G$.