Section Three.
Hypothesis alpha_irr : forall n p : Z, (alpha * IZR p)%R <> IZR n.
Hypothesis prop_alpha : (0 < alpha)%R /\ (alpha < 1)%R.
Hypothesis prop_N : forall N : nat, N >= 2.

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Lemma three_gap1 :
forall N n : nat, 0 < n -> n < N - first N -> after N n = n + first N.