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,u:A][t1:(R x u)][t2:(Rstar u y)][v:A][u1:(R x v)][u2:(Rstar v z)]
([Hyp0:(x0,y0,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,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)]
[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).

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