intro.
cut (derivable_pt (fun x : R => (vi intr * sin (thetat x - alphas x))%R) t).
intro X.
replace
(derive_pt (fun y : R => (Rs y * derive_pt alphas y (cond_D1 alphas y))%R) t
(fct_der5 t)) with
(derive_pt (fun x : R => (vi intr * sin (thetat x - alphas x))%R) t X).
set (f := d2 alphas).
cut (derivable_pt f t); [ intro X0 | apply (cond_D1 alphas) ].
cut (derivable_pt thetat t); [ intro X1 | apply thetat_derivable ].
reg.
unfold fct_cte, minus_fct in |- *.
replace (derive_pt f t (cond_D1 alphas t)) with (derive_pt f t X0);
[ idtac | apply pr_nu ].
replace (derive_pt thetat t (thetat_derivable t)) with
(derive_pt thetat t X1); [ idtac | apply pr_nu ].
ring.
unfold derive_pt in |- *.
case X; case (fct_der5 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 H);
intros.
exists x1; intros.
do 2 rewrite <- Rs_alphas.
apply H0; assumption.
fold alphas_p in |- *; reg.
apply (cond_D1 alphas).
apply thetat_derivable.
Qed.

(* Specifications: *)
(**********)
Lemma D_Rs_2 :
forall t : R,
(derive_pt Rs t (Rs_derivable t) *
(2 * derive_pt alphas t (cond_D1 alphas t) -
derive_pt thetat t (thetat_derivable t)))%R =
(- Rs t * derive_pt (derive alphas (cond_D1 alphas)) t (cond_D2 alphas t))%R.