Axiom A : Prop.Axiom R : A->A->Prop.Axiom Rstar: A->A->Prop.Axiom Rstar': A->A->Prop.Axiom coherence: A->A->Prop.Axiom Hyp1:(x:A)(P:A->Prop)((y:A)((z:A)(R y z)->(P z))->(P y))->(P x).Axiom Hyp2:(x:A)(y:A)(z:A)(R x y)->(R x z)->(coherence y z).

Theorem Ind_proof : (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)->(coherence y z).