intro.
cut
(derive_pt G t (G_derivable t) =
(derive_pt Rs t (Rs_derivable t) - derive_pt F t (F_derivable t))%R).
intro; rewrite H;
rewrite
(Rsqr_minus_plus (derive_pt Rs t (Rs_derivable t))
(derive_pt F t (F_derivable t))); unfold G in |- *;
repeat rewrite Rmult_assoc; rewrite (Rsqr_minus_plus (Rs t) (F t)).
replace
(- Rsqr (derive_pt alphas t (cond_D1 alphas t)) * (Rsqr (Rs t) - Rsqr (F t)) +
Rsqr (F t) *
(Rsqr (rho (vi intr) / 2) - Rsqr (derive_pt alphas t (cond_D1 alphas t))))%R
with
(- Rsqr (derive_pt alphas t (cond_D1 alphas t)) * Rsqr (Rs t) +
Rsqr (F t) * Rsqr (rho (vi intr) / 2))%R.
apply
Rplus_eq_reg_l
with
(Rsqr (derive_pt F t (F_derivable t)) +
Rsqr (derive_pt alphas t (cond_D1 alphas t)) * Rsqr (Rs t))%R.
replace
(Rsqr (derive_pt F t (F_derivable t)) +
Rsqr (derive_pt alphas t (cond_D1 alphas t)) * Rsqr (Rs t) +
(Rsqr (derive_pt Rs t (Rs_derivable t)) -
Rsqr (derive_pt F t (F_derivable t))))%R with
(Rsqr (derive_pt Rs t (Rs_derivable t)) +
Rsqr (Rs t) * Rsqr (derive_pt alphas t (cond_D1 alphas t)))%R.
replace
(Rsqr (derive_pt F t (F_derivable t)) +
Rsqr (derive_pt alphas t (cond_D1 alphas t)) * Rsqr (Rs t) +
(- Rsqr (derive_pt alphas t (cond_D1 alphas t)) * Rsqr (Rs t) +
Rsqr (F t) * Rsqr (rho (vi intr) / 2)))%R with
(Rsqr (derive_pt F t (F_derivable t)) +
Rsqr (rho (vi intr) / 2) * Rsqr (F t))%R.
rewrite Rsqr_DRs; rewrite D_F_Rsqr_v; reflexivity.
ring.
ring.
ring.
unfold G in |- *.
cut (derivable_pt Rs t); [ intro X | apply Rs_derivable ].
cut (derivable_pt F t); [ intro X0 | apply F_derivable ].
reg.
replace (derive_pt F t (F_derivable 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 Rlt_1_PI2 : (1 < PI / 2)%R.
generalize PI_approx; intro; elim H; intros.