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).