Proof with trivial.
generalize (dtheta_rho intr); intros; split...
unfold thetat in |- *; cut (derivable (fun y : R => (- rho_vi * y)%R));
[ intro X | reg ]...
cut
(forall z : R,
derive_pt (fun y : R => (- rho_vi * y)%R) z (X z) = (- rho_vi)%R);
[ intro | intro; reg ]...
apply Rplus_le_reg_l with (- theta intr 0)%R...
replace (- theta intr 0 + (theta intr 0 - rho_vi * t))%R with
(- rho_vi * t - - rho_vi * 0)%R; [ idtac | ring ]...
replace (- theta intr 0 + theta intr t)%R with
(theta intr t - theta intr 0)%R; [ idtac | ring ]...
apply
(IAF_var (theta intr) (fun y : R => (- rho_vi * y)%R) 0 t
(cond_diff (theta intr)) X H0)...
intros; rewrite (H1 c); unfold rho_vi in |- *; unfold vi in |- *; elim (H c);
intros H4 _; assumption...
unfold thetat in |- *; cut (derivable (fun y : R => (rho_vi * y)%R));
[ intro X | reg ]...
cut (forall z : R, derive_pt (fun y : R => (rho_vi * y)%R) z (X z) = rho_vi);
[ intro | intro; reg ]...
apply Rplus_le_reg_l with (- theta intr 0)%R...
replace (- theta intr 0 + (theta intr 0 + rho_vi * t))%R with
(rho_vi * t - rho_vi * 0)%R; [ idtac | ring ]...
replace (- theta intr 0 + theta intr t)%R with
(theta intr t - theta intr 0)%R; [ idtac | ring ]...
apply
(IAF_var (fun y : R => (rho_vi * y)%R) (theta intr) 0 t X
(cond_diff (theta intr)) H0)...
intros; rewrite (H1 c); unfold rho_vi in |- *; unfold vi in |- *; elim (H c)...
Qed.

Lemma rho_t_pos : forall t : R, (0 <= t)%R -> (0 <= rho_vi * t)%R.
Proof with trivial.