Definition thetat : R -> R := theta intr.Definition Rs (t : R) : R :=
  sqrt (Rsqr (xi intr t - xi intr 0) + Rsqr (yi intr t - yi intr 0)).Axiom
  alphas_exists :
    { f : Differential_D2 |
       forall t : R,
       xi intr t = (Rs t * cos (f t) + xi intr 0)%R /\
       yi intr t = (Rs t * sin (f t) + yi intr 0)%R }.Definition alphas := let (a,_) := alphas_exists in a.

(* Specifications: *)
Lemma alphas_def : forall t : R, xi intr t = (Rs t * cos (alphas t) + xi intr 0)%R /\ yi intr t = (Rs t * sin (alphas t) + yi intr 0)%R.
Lemma Rsx : forall t : R, xi intr t = (xi intr 0 + Rs t * cos (alphas t))%R.
Lemma Rsy : forall t : R, yi intr t = (yi intr 0 + Rs t * sin (alphas t))%R.
Lemma Rs_x : forall t : R, cos (alphas t) <> 0%R -> Rs t = ((xi intr t - xi intr 0) / cos (alphas t))%R.
Lemma Rs_y : forall t : R, sin (alphas t) <> 0%R -> Rs t = ((yi intr t - yi intr 0) / sin (alphas t))%R.
Lemma Rs_derivable : derivable Rs.