intro.
assert (H := fct_der4).
replace (derive_pt (xi intr) t (xi_derivable intr t)) with
(derive_pt ((Rs * comp cos alphas_p)%F + fct_cte (xi intr 0)) t (H t)).
unfold plus_fct, mult_fct, comp, fct_cte in |- *.
set (H0 := Rs_derivable t).
cut (derivable_pt alphas_p t); [ intro X | apply (cond_D1 alphas) ].
replace (derive_pt alphas t (cond_D1 alphas t)) with (derive_pt alphas_p t X);
[ idtac | apply pr_nu ].
reg.
unfold comp, alphas_p in |- *; ring.
unfold derive_pt in |- *.
case (H t).
case (xi_derivable intr t).
simpl.
intros.
eapply uniqueness_limite.
unfold derivable_pt_abs in d0.
apply d0.
unfold derivable_pt_lim in |- *; intros.
unfold derivable_pt_abs in d; unfold derivable_pt_lim in d.
elim (d eps H0); intros.
exists x1; intros.
unfold plus_fct, mult_fct, comp, fct_cte in |- *.
assert (H4 := H1 _ H2 H3).
replace
((Rs (t + h) * cos (alphas_p (t + h)) + xi intr 0 -
(Rs t * cos (alphas_p t) + xi intr 0)) / h - x)%R with
((xi intr (t + h) - xi intr t) / h - x)%R.
apply H4.
rewrite (Rsx t).
rewrite (Rsx (t + h)).
replace
(xi intr 0 + Rs (t + h) * cos (alphas (t + h)) -
(xi intr 0 + Rs t * cos (alphas t)))%R with
(Rs (t + h) * cos (alphas_p (t + h)) + xi intr 0 -
(Rs t * cos (alphas_p t) + xi intr 0))%R.
reflexivity.
unfold alphas_p in |- *; ring.
Qed.

(* Specifications: *)
Lemma Dyi :
forall t : R,
derive_pt (yi intr) t (yi_derivable intr t) =
(derive_pt Rs t (Rs_derivable t) * sin (alphas t) +
Rs t * cos (alphas t) * derive_pt alphas t (cond_D1 alphas t))%R.