Does this proof in fact use induction?

[Genady]

Consider the following proof:

My question is, does it in fact use induction?

It says, "Assume now that the theorem is true for k-1 elements," but I don't see this assumption being used in the proof to advance from k-1 to k elements, which would be an induction step. Actually, this assumption is not used at all, AFAICS.

Edited December 16 by Genady