Qed.

(**********)
Lemma r_lb_0 : forall h : TypeSpeed, (r_lb h < r h)%R.
intros; unfold r, r_lb in |- *; generalize (rho_ub_0 h); intro H;
generalize (Rmult_lt_0_compat (rho h) (rho_ub h) (rho_pos h) (rho_ub_pos h));
intro H0; generalize (Rinv_lt_contravar (rho h) (rho_ub h) H0 H);
intro H1;
generalize (Rmult_lt_compat_l h (/ rho_ub h) (/ rho h) (TypeSpeed_pos h) H1);
intro H2; assumption.