Show that there is no integer between 8 and 7 by using the statement (not the proof) of the Lemma 1.1: ”There is no integer between 0 and 1”.

By the contrary: if the integer x exists, than 7<x<8, and x-integer. So, y=x-7 is also integer and 0<y<1. It is contradiction of the Lemma statement

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