Definition map_pair {A B C D} (f:A -> B) (g:C -> D) (x:A * C) : B * D :=
  (f (fst x), g (snd x)).Module TF := TF'.Module K1 := K1'.Module K2 := K2'.Module K1L := Knot_Lemmas K1.Module K2L := Knot_Lemmas K2.Definition f_pred' (p1 : K1.predicate) : K2.predicate :=
  p1 oo map_pair g (@id other).Definition g_pred' (p2 : K2.predicate) : K1.predicate :=
  p2 oo map_pair f (@id other).

Lemma f_pred'_approx: forall n, f_pred' oo K1.approx n = K2.approx n oo f_pred'.