## Basis extension

Prove that if $$V$$ is a subspace of a finitely generated space $$W$$ then there exist bases $$X$$ and $$Y$$, resp., of $$V$$ and $$W$$, resp., such that $$X \subseteq Y$$.
Choose any basis $$X$$ of $$V$$ and any auxiliary basis $$Y'$$ of $$W$$.
By the exchange theorem , ($$X$$ is independent and $$Y'$$ generates $$W$$) we find $$Y$$ such that $$X\subseteq Y$$, $$Y$$ generates $$W$$ and $$|Y|=|Y'|$$. From the last two conditions follows that $$Y$$ is the desired basis of $$W$$.