Proof with trivial.
intros; split...
apply (IAF_var yp hy 0 t yp_derivable hy_derivable H)...
intros; elim H1; intros...
generalize
(Dyp0_PI2 c H2
(Rle_trans (rho_vi * c) (rho_vi * t) (PI / 2) (rho_t_le c t H3) H0));
intro...
elim H4; intros...
cut (derive_pt hy c (hy_derivable c) = (- (vi intr * sin (rho_vi * c)))%R)...
intro...
rewrite <- H7 in H5...
rewrite Dhy; ring...
replace (hy 0 - hy t)%R with ((- hy)%F t - (- hy)%F 0)%R...
apply
(IAF_var (- hy)%F yp 0 t (derivable_opp hy hy_derivable) yp_derivable H)...
intros; elim H1; intros...
generalize
(Dyp0_PI2 c H2
(Rle_trans (rho_vi * c) (rho_vi * t) (PI / 2) (rho_t_le c t H3) H0));
intro...
elim H4; intros...
cut ((- derive_pt hy c (hy_derivable c))%R = (vi intr * sin (rho_vi * c))%R)...
intro; rewrite <- H7 in H6...
replace (derive_pt (- hy) c (derivable_opp hy hy_derivable c)) with
(derive_pt (- hy) c (derivable_pt_opp hy c (hy_derivable c)));
[ idtac | apply pr_nu ]...
rewrite (derive_pt_opp hy c (hy_derivable c))...
rewrite Dhy; ring...
unfold opp_fct in |- *; ring...
Qed.

Lemma ypt_PI2 :
forall t : R,
(0 <= t)%R ->
(rho_vi * t <= PI / 2)%R ->
(l * sin (beta + thetat intr 0) + r_vi * (cos (rho_vi * t) - 1) <= yp t)%R /\
(yp t <= l * sin (beta + thetat intr 0) - r_vi * (cos (rho_vi * t) - 1))%R.