intro; unfold yp in |- *.
assert (H := xi_derivable intr t).
assert (H0 := yi_derivable intr t).
assert (H1 := thetat_derivable t).
set (xi_p := xi intr) in H |- *.
set (yi_p := yi intr) in H0 |- *.
set (thetat_p := thetat intr) in H1 |- *.
reg.
assert (H2 := cond_x intr t).
assert (H3 := cond_y intr t).
replace (derive_pt xi_p t H) with
(derive_pt (x intr) t (cond_diff (x intr) t)); [ idtac | apply pr_nu ].
replace (derive_pt yi_p t H0) with
(derive_pt (y intr) t (cond_diff (y intr) t)); [ idtac | apply pr_nu ].
rewrite H2; rewrite H3.
rewrite sin_minus.
unfold fct_cte, vi, thetat_p, thetat in |- *.
ring.
Qed.

(* Specifications: *)
Lemma Dyp0_PI2 :
forall t : R,
(0 <= t)%R ->
(rho_vi * t <= PI / 2)%R ->
(- (vi intr * sin (rho_vi * t)) <= derive_pt yp t (yp_derivable t) <=
vi intr * sin (rho_vi * t))%R.