Lemma cF_L1:forall(m:fmap)(x y z:dart), inv_hmap m -> prec_L m one x y -> let x10 := cF_1 m x in let y0 := cA m zero y in let y_1 := cA_1 m one y in let m1:= L m one x y in cF m1 z = if eq_dart_dec y0 z then x else if eq_dart_dec x10 z then y_1 else cF m z.