intro.
set (H := Rs_derivable t).
cut (derivable_pt alphas_p t); [ intro X | apply (cond_D1 alphas) ].
cut (derivable_pt ((Rs * comp sin alphas_p)%F + fct_cte (yi intr 0)) t);
[ intro X0 | reg ].
replace (derive_pt (yi intr) t (yi_derivable intr t)) with
(derive_pt ((Rs * comp sin alphas_p)%F + fct_cte (yi intr 0)) t X0).
replace (derive_pt alphas t (cond_D1 alphas t)) with (derive_pt alphas_p t X);
[ idtac | apply pr_nu ].
unfold plus_fct, mult_fct, comp, fct_cte in |- *.
reg.
unfold comp, alphas_p in |- *; ring.
unfold derive_pt in |- *.
case X0.
case (yi_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) * sin (alphas_p (t + h)) + yi intr 0 -
(Rs t * sin (alphas_p t) + yi intr 0)) / h - x)%R with
((yi intr (t + h) - yi intr t) / h - x)%R.
apply H4.
rewrite (Rsy t).
rewrite (Rsy (t + h)).
replace
(yi intr 0 + Rs (t + h) * sin (alphas (t + h)) -
(yi intr 0 + Rs t * sin (alphas t)))%R with
(Rs (t + h) * sin (alphas_p (t + h)) + yi intr 0 -
(Rs t * sin (alphas_p t) + yi intr 0))%R.
reflexivity.
unfold alphas_p in |- *; ring.
Qed.

(* Specifications: *)
(**********)
Lemma DRs_cos :
forall t : R,
(derive_pt Rs t (Rs_derivable t) * cos (alphas t) -
Rs t * sin (alphas t) * derive_pt alphas t (cond_D1 alphas t))%R =
(vi intr * cos (thetat t))%R.