Attached is a proof of the theorem.

I don't understand how he comes to the conclusion in the inductive step that B and C "can only differ in the nth column". I understand that any operations that put A in rre form put A' in rre form. I also understand the concept and method of induction, I'm just not seeing it in this instance. I don't see why they can't differ in columns 1 through (n-1).