I'm really stuck on the proof of this theorem...

Replacement Theorem: Let

be a vector space that is generated by a set

containing exactly

elements, and let

be a linearly independent subset of

containing exactly

elements. Then

and there exists a subset

of

containing exactly

elements such that

generates

.

Proof. The proof is by induction on

. The induction begins with

; for in this case

, and so taking

gives the desired result.

Now suppose the theorem is true for some integer

. We prove that the theorem is true for

. Let

be a linearly independent subset of

consisting of

elements. By the corollary to Theorem 1.6 (if

and

is linearly independent, then

is linearly independent)

is linearly independent, and so we may apply the induction hypothesis to conclude that

and that there is a subset

of

such that

generates

. ... (from Linear Algebra, 3rd ed., Friedberg et al.)

How exactly is the induction hypothesis being used? I'm really missing something.