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:(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)]
(h Rstar' [u:A](Rstar'_reflexive u)
[u,v,w:A][h1:(R u v)]
[h2:(Rstar' v w)](Rstar'_R u w v h1 (Rstar'_Rstar v w h2))).