Definition f_Sn (k1 : K1.knot) : K2.knot :=
  match K1.unsquash k1 with
    (n, F_p1) => K2.squash (n, f_F' g F_p1)
  end.Definition g_Sn (k2 : K2.knot) : K1.knot :=
  match K2.unsquash k2 with
    (n, F_p2) => K1.squash (n, g_F' f F_p2)
  end.

Lemma gn_unsquash : forall k2, level k2 <= n -> forall U2, U2 = K2.unsquash k2 -> K1.unsquash (g k2) = (fst U2, g_F' f (snd U2)).