Variable A : Prop .
Variable R : A->A->Prop.
Definition Rstar:A->A->Prop :=
[x,y:A](P0:A->A->Prop)
((u:A)(P0 u u))
->((u:A)(v:A)(w:A)(R u v)->(P0 v w)->(P0 u w))->(P0 x y).
Definition Rstar':A->A->Prop :=
[x,y:A](P:A->A->Prop)
(P x x)->((u:A)(R x u)->(Rstar u y)->(P x y))->(P x y).
Definition coherence: A->A->Prop :=
[x:A][y:A](P:Prop)((z:A)(Rstar x z)->(Rstar y z)->P)->P.
(* Les 2 hypotheses du Lemme de Newman *)

Lemma Rstar'_Rstar:  (x:A)(y:A)(Rstar' x y)->(Rstar x y).
Proof [x,y:A][h:(P:A->A->Prop)
(P x x)->((u:A)(R x u)->(Rstar u y)->(P x y))->(P x y)]
(h [a,a0:A](Rstar a a0) (Rstar_reflexive x) [u:A](Rstar_R x u y)).