End FG.
(* Now for the main definitions and theorems *)
Definition f (k : K1.knot) : K2.knot := fst (fg (level k)) k.
Definition g (k : K2.knot) : K1.knot := snd (fg (level k)) k.
Definition fF : TF.F K1.predicate -> TF.F K2.predicate :=
f_F' g.
Definition gF : TF.F K2.predicate -> TF.F K1.predicate :=
g_F' f.

Lemma f_level : forall k, level k = level (f k).