intro; rewrite <- H in H0;
cut (frac_part (INR 0 * alpha) = frac_part_n_alpha 0).
intro; rewrite H1 in H0; generalize (tech_fp_alp_irr 0 (first N) H0); intro;
generalize (first_0 N (prop_N N)); intro;
generalize (lt_not_eq 0 (first N) H3); intro; elimtype False;
auto.
unfold frac_part_n_alpha in |- *; auto.
auto.
Qed.

(**********)
Lemma contra_tech_fp_alp_irr :
forall n m : nat, n <> m -> frac_part_n_alpha n <> frac_part_n_alpha m.