intro.
set (f := fun x : R => derive_pt alphas x (cond_D1 alphas 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 y : R => (Rs y * f y)%R) t (fct_der5 t));
[ idtac | reflexivity ].
replace (derive_pt alphas t (cond_D1 alphas t)) with (f t);
[ idtac | reflexivity ].
replace (derive_pt (derive alphas (cond_D1 alphas)) t (cond_D2 alphas t))
with (derive_pt f t (cond_D2 alphas t)); [ idtac | reflexivity ].
cut (derivable_pt Rs t); [ intro X | apply Rs_derivable ].
cut (derivable_pt f t); [ intro X0 | apply (cond_D2 alphas) ].
reg.
replace (derive_pt f t (cond_D2 alphas t)) with (derive_pt f t X0);
[ idtac | apply pr_nu ].
replace (derive_pt Rs t (Rs_derivable t)) with (derive_pt Rs t X);
[ idtac | apply pr_nu ].
ring.
Qed.

(**********)
Lemma thetat_derivable : derivable thetat.
unfold thetat in |- *; apply (cond_diff (theta intr)).