help needed in Numbers Game

Numbers Game.
looking at editorial i can’t understand this line from the editorial: “Assuming that in all steps (x−q∗y) ≥ (r∗y) " , how we can prove that we can always have this condition what if” (x−q∗y) < (r∗y) " ,
how can we say these two conditions are interchangeable , if possible provide a proof, i tried hard to prove but can"t reach . please help

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