intro.
assert (H := D_Rs_alphas1 t).
assert (H0 := D_Rs_alphas2 t).
assert (H1 := DRs t).
rewrite H in H0; rewrite <- H1 in H0.
assert
(H2 :=
Rplus_eq_compat_l
(- derive_pt Rs t (Rs_derivable t) *
(derive_pt thetat t (thetat_derivable t) -
derive_pt alphas t (cond_D1 alphas t)) +
- Rs t * derive_pt (derive alphas (cond_D1 alphas)) t (cond_D2 alphas t))
_ _ H0).
cut
((- derive_pt Rs t (Rs_derivable t) *
(derive_pt thetat t (thetat_derivable t) -
derive_pt alphas t (cond_D1 alphas t)) +
- Rs t * derive_pt (derive alphas (cond_D1 alphas)) t (cond_D2 alphas t) +
(derive_pt Rs t (Rs_derivable t) * derive_pt alphas t (cond_D1 alphas t) +
Rs t * derive_pt (derive alphas (cond_D1 alphas)) t (cond_D2 alphas 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); [ intro | ring ].
cut
((- derive_pt Rs t (Rs_derivable t) *
(derive_pt thetat t (thetat_derivable t) -
derive_pt alphas t (cond_D1 alphas t)) +
- Rs t * derive_pt (derive alphas (cond_D1 alphas)) t (cond_D2 alphas t) +
derive_pt Rs t (Rs_derivable t) *
(derive_pt thetat t (thetat_derivable t) -
derive_pt alphas t (cond_D1 alphas t)))%R =
(- Rs t * derive_pt (derive alphas (cond_D1 alphas)) t (cond_D2 alphas t))%R);
[ intro | ring ].
rewrite H3 in H2; rewrite H4 in H2.
assumption.
Qed.

(**********)
Lemma Rsqr_DRs :
forall t : R,
(Rsqr (derive_pt Rs t (Rs_derivable t)) +
Rsqr (Rs t) * Rsqr (derive_pt alphas t (cond_D1 alphas t)))%R =
Rsqr (vi intr).