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+Axiom fun_extensionality : forall A B (f g : A -> B), 
+  (forall x, f x = g x) -> f = g.
+
+Axiom fun_extensionality_dep : forall A, forall B : (A -> Type), forall (f g : forall x : A, B x), 
+  (forall x, f x = g x) -> f = g.
+
+Hint Resolve fun_extensionality fun_extensionality_dep : subtac.
+
+Require Import Coq.subtac.Utils.
+Require Import Coq.subtac.FixSub.
+
+Lemma fix_sub_eq_ext :
+  forall (A : Set) (R : A -> A -> Prop) (Rwf : well_founded R)
+    (P : A -> Set)
+    (F_sub : forall x : A, (forall  {y : A | R y x}, P (`y)) -> P x),
+    forall x : A,
+      Fix_sub A R Rwf P F_sub x =
+        F_sub x (fun {y : A | R y x}=> Fix A R Rwf P F_sub (`y)).
+Proof.
+  intros ; apply Fix_eq ; auto.
+  intros.
+  assert(f = g).
+  apply (fun_extensionality_dep _ _ _ _ H).
+  rewrite H0 ; auto.
+Qed.