Definition f_F' : TF.F K1.predicate -> TF.F K2.predicate :=
  fmap F f_pred'.Definition g_F' : TF.F K2.predicate -> TF.F K1.predicate :=
  fmap F g_pred'.(* The base case.  To keep things simple, we will put definitions first. *)

Definition fZ_pred (p2 : K2.predicate) : K1.predicate :=
fun k1 => T_bot.Definition gZ_pred (p1 : K1.predicate) : K2.predicate :=
fun k2 => T_bot.Definition fF_Z : TF.F K1.predicate -> TF.F K2.predicate :=
  fmap F gZ_pred.Definition gF_Z : TF.F K2.predicate -> TF.F K1.predicate :=
  fmap F fZ_pred.Definition f_Z (k1 : K1.knot) : K2.knot :=
  match K1.unsquash k1 with
    (n, F_p1) => K2.squash (n, fF_Z F_p1)
  end.Definition g_Z (k2 : K2.knot) : K1.knot :=
  match K2.unsquash k2 with
    (n, F_p2) => K1.squash (n, gF_Z F_p2)
  end.

Lemma g_level_Z: forall k, level k = level (g_Z k).