Proof [x:A]
(unf_Rstar [P:A->A->Prop](P x x)
[P0:A->A->Prop]
[H:(u:A)(P0 u u)][_:(u:A)(v:A)(w:A)(R u v)->(P0 v w)->(P0 u w)](H x)).
Proof [x,y:A]
[z:A]
[t1:(R x y)]
(unf_Rstar [P:A->A->Prop](P y z)->(P x z)
[t2:(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 y z)]
[P:A->A->Prop]
[h1:(u:A)(P u u)]
[h2:(u:A)(v:A)(w:A)(R u v)->(P v w)->(P u w)]
(h2 x y z t1 (t2 [a,a0:A](P a a0) h1 h2))).
Proof [x,y:A]
[z:A]
(unf_Rstar [P:A->A->Prop](P x y)->(Rstar y z)->(Rstar x z)
[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 [a,a0:A](Rstar a0 z)->(Rstar a z) [u:A][H:(Rstar u z)]H
[u,v:A]
[w:A]
[t1:(R u v)]
[t2:(Rstar w z)->(Rstar v z)]
[t3:(Rstar w z)](Rstar_R u v z t1 (t2 t3)))).
Proof [x:A]
(unf_Rstar' [P:A->A->Prop](P x x)
[P:A->A->Prop][H:(P x x)][_:(u:A)(R x u)->(Rstar u x)->(P x x)]H).

Theorem Rstar'_R: (x:A)(y:A)(z:A)(R x z)->(Rstar z y)->(Rstar' x y).