Proof with trivial.
intros; apply Rlt_le_trans with (Rsqr (xp t))...
apply Rsqr_incrst_1...
apply (no_conflict_xp_1_Omega t H H0 H1)...
unfold ConflictRange in |- *; left; prove_sup...
cut ((0 <= t)%R /\ (rho_vi * t <= PI)%R)...
intro; elim H2; intros; apply (xp_pos t H3 H4 H1)...
split...
left; apply Rlt_le_trans with 1%R; try apply Rlt_0_1...
unfold Rminus in H0;
generalize (Rplus_le_compat_r rho_vi (rho_vi * t) (PI + - rho_vi) H0);
repeat rewrite Rplus_assoc; rewrite Rplus_opp_l; rewrite Rplus_0_r;
intro; apply (plus_le_is_le (rho_vi * t) rho_vi PI)...
unfold rho_vi in |- *; left; apply rho_pos...
elim (majoration t)...
Qed.

Lemma no_conflict_scenario_0_1s :
forall t : R,
(0 <= t)%R ->
(t < 1)%R -> (ConflictRange + vi intr < l)%R -> (ConflictRange < e t)%R.