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Does this proof in fact use induction?

Genady
in Linear Algebra and Group Theory

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Consider the following proof:

image.png.3535a9a07e6a1ae063cf7e5cd220239c.png

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 by Genady

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