Definition alphas_p : R -> R := d2 alphas.

(* Specifications: *)
Lemma fct_der4 : forall t : R, derivable_pt ((Rs * comp cos alphas_p)%F + fct_cte (xi intr 0)) t.
Lemma Dxi : forall t : R, derive_pt (xi intr) t (xi_derivable intr t) = (derive_pt Rs t (Rs_derivable t) * cos (alphas t) - Rs t * sin (alphas t) * derive_pt alphas t (cond_D1 alphas t))%R.
Lemma Dyi : forall t : R, derive_pt (yi intr) t (yi_derivable intr t) = (derive_pt Rs t (Rs_derivable t) * sin (alphas t) + Rs t * cos (alphas t) * derive_pt alphas t (cond_D1 alphas t))%R.
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.
Lemma DRs_sin : forall t : R, (derive_pt Rs t (Rs_derivable t) * sin (alphas t) + Rs t * cos (alphas t) * derive_pt alphas t (cond_D1 alphas t))%R = (vi intr * sin (thetat t))%R.
Lemma eq_plus_eq_is_eq : forall x y z t : R, x = y -> z = t -> (x + z)%R = (y + t)%R.
Lemma DRs : forall t : R, derive_pt Rs t (Rs_derivable t) = (vi intr * cos (thetat t - alphas t))%R.
Lemma Rs_alphas : forall t : R, (Rs t * derive_pt alphas t (cond_D1 alphas t))%R = (vi intr * sin (thetat t - alphas t))%R.
Lemma fct_der5 : forall t : R, derivable_pt (fun y : R => (Rs y * derive_pt alphas y (cond_D1 alphas y))%R) t.
Lemma D_Rs_alphas1 : forall t : R, derive_pt (fun y : R => (Rs y * derive_pt alphas y (cond_D1 alphas y))%R) t (fct_der5 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.
Lemma thetat_derivable : derivable thetat.
Lemma D_Rs_alphas2 : forall t : R, derive_pt (fun y : R => (Rs y * derive_pt alphas y (cond_D1 alphas y))%R) t (fct_der5 t) = (vi intr * cos (thetat t - alphas t) * (derive_pt thetat t (thetat_derivable t) - derive_pt alphas t (cond_D1 alphas t)))%R.
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.
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).
Lemma Rs_pos : forall t : R, (0 <= Rs t)%R.
Lemma Rs_0 : Rs 0 = 0%R.
Lemma D_Rs_0 : (0 <= derive_pt Rs 0 (Rs_derivable 0))%R.
Lemma eq_sym : forall x y : R, x = y -> y = x.
Lemma D_Rs_0_strong : (0 < derive_pt Rs 0 (Rs_derivable 0))%R.
Lemma D_alphas_0 : (2 * derive_pt alphas 0 (cond_D1 alphas 0))%R = derive_pt thetat 0 (thetat_derivable 0%R).
Lemma D_Rs_pos_non_null : forall t : R, (0 < t)%R -> (forall x : R, (0 < x < t)%R -> (0 < derive_pt Rs x (Rs_derivable x))%R) -> (0 < Rs t)%R.