Section gap.
Variable N n : nat.
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.
Hypothesis Hn : 0 < n /\ n < N.
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Definition succes := after N n.
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Definition num1 := n + first N.
Definition num2 := n + first N - last N.
Definition num3 := n - last N.
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(*                 Theorem                               *)
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Theorem three_gap : succes = num1 \/ succes = num2 \/ succes = num3.