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).
Proof [x,y:A]
[z:A]
[t1:(R x z)]
[t2:(Rstar z y)]
(unf_Rstar' [P:A->A->Prop](P x y)
[P:A->A->Prop]
[_:(P x x)][h2:(u:A)(R x u)->(Rstar u y)->(P x y)](h2 z t1 t2)).
Proof [x,y:A]
(unf_Rstar' [P:A->A->Prop](P x y)->(Rstar x y)
[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))).
Proof  [x,y:A]
(unf_Rstar [P:A->A->Prop](P x y)->(Rstar' x y)
[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:A]
[w:A]
[h1:(R u v)]
[h2:(Rstar' v w)](Rstar'_R u w v h1 (Rstar'_Rstar v w h2)))).
Proof [x,y:A]
[z:A]
[H:(Rstar x z)]
[H0:(Rstar y z)]
(unf_coherence [P:A->A->Prop](P x y)
[P:Prop][H1:(z0:A)(Rstar x z0)->(Rstar y z0)->P](H1 z H H0)).
Proof  [x:A][y:A][h:(Rstar x y)](coherence_intro x y y h (Rstar_reflexive y)).
Proof [x,y:A]
(unf_coherence [P:A->A->Prop](P x y)->(P y x)
[H:(P:Prop)((z:A)(Rstar x z)->(Rstar y z)->P)->P]
[P:Prop]
[H0:(z:A)(Rstar y z)->(Rstar x z)->P]
(H P [z:A][H1:(Rstar x z)][H2:(Rstar y z)](H0 z H2 H1))).
Proof [x:A]
[hyp_ind:(u:A)
(R x u)->(y:A)(z:A)(Rstar u y)->(Rstar u z)->(coherence y z)]
[y,z:A]
[u:A]
[t1:(R x u)]
[t2:(Rstar u y)]
[v:A]
[u1:(R x v)]
[u2:(Rstar v z)]
(unf_coherence
[P:A->A->Prop]
((x0:A)(y0:A)(z0:A)(R x0 y0)->(R x0 z0)->(P y0 z0))
->((u0:A)
(R x u0)
->(y0:A)(z0:A)(Rstar u0 y0)->(Rstar u0 z0)->(P y0 z0))
->(coherence y z)
[Hyp0:(x0:A)
(y0:A)
(z0:A)
(R x0 y0)
->(R x0 z0)
->(P:Prop)
((z1:A)(Rstar y0 z1)->(Rstar z0 z1)->P)->P]
[hyp_ind0:(u0:A)
(R x u0)
->(y0:A)
(z0:A)
(Rstar u0 y0)
->(Rstar u0 z0)
->(P:Prop)
((z1:A)
(Rstar y0 z1)->(Rstar z0 z1)->P)
->P]
(Hyp0 x u v t1 u1 (coherence y z)
[z0:A]
[H:(Rstar u z0)]
[H0:(Rstar v z0)]
(hyp_ind0 u t1 y z0 t2 H (coherence y z)
[z1:A]
[H1:(Rstar y z1)]
[H2:(Rstar z0 z1)]
(hyp_ind0 v u1 z z1 u2
(Rstar_transitive v z0 z1 H0 H2)
(coherence y z)
[z2:A]
[H3:(Rstar z z2)]
[H4:(Rstar z1 z2)]
(unf_coherence [P:A->A->Prop](P y z)
[P:Prop]
[H5:(z3:A)(Rstar y z3)->(Rstar z z3)->P]
(H5 z2
(Rstar_transitive y z1 z2 H1 H4) H3)))))
Hyp2 hyp_ind).

Theorem caseRxy: (x:A)
((u:A)
(R x u)
->(y:A)(z:A)(Rstar u y)->(Rstar u z)->(coherence y z))
->(y:A)
(z:A)
(Rstar x y)
->(Rstar x z)
->(u:A)(R x u)->(Rstar u y)->(coherence y z).