Definition Carrier := Ensemble U.Definition Rel := Relation U.Record PO : Type := Definition_of_PO
    { Carrier_of : Ensemble U;
      Rel_of : Relation U;
      PO_cond1 : Inhabited U Carrier_of;
      PO_cond2 : Order U Rel_of }.Record PO : Type := Definition_of_PO
    { Carrier_of : Ensemble U;
      Rel_of : Relation U;
      PO_cond1 : Inhabited U Carrier_of;
      PO_cond2 : Order U Rel_of }.Record PO : Type := Definition_of_PO
    { Carrier_of : Ensemble U;
      Rel_of : Relation U;
      PO_cond1 : Inhabited U Carrier_of;
      PO_cond2 : Order U Rel_of }.Record PO : Type := Definition_of_PO
    { Carrier_of : Ensemble U;
      Rel_of : Relation U;
      PO_cond1 : Inhabited U Carrier_of;
      PO_cond2 : Order U Rel_of }.Record PO : Type := Definition_of_PO
    { Carrier_of : Ensemble U;
      Rel_of : Relation U;
      PO_cond1 : Inhabited U Carrier_of;
      PO_cond2 : Order U Rel_of }.Record PO : Type := Definition_of_PO
    { Carrier_of : Ensemble U;
      Rel_of : Relation U;
      PO_cond1 : Inhabited U Carrier_of;
      PO_cond2 : Order U Rel_of }.Definition Strict_Rel_of : Rel := fun x y:U => Rel_of p x y /\ x <> y.Inductive covers (y x:U) : Prop :=
    Definition_of_covers :
    Strict_Rel_of x y ->
    ~ (exists z : _, Strict_Rel_of x z /\ Strict_Rel_of z y) ->
    covers y x.Inductive covers (y x:U) : Prop :=
    Definition_of_covers :
    Strict_Rel_of x y ->
    ~ (exists z : _, Strict_Rel_of x z /\ Strict_Rel_of z y) ->
    covers y x.

(* Specifications: *)
Lemma Strict_Rel_Transitive_with_Rel : forall x y z:U, Strict_Rel_of U D x y -> @Rel_of U D y z -> Strict_Rel_of U D x z.
Lemma Strict_Rel_Transitive_with_Rel_left : forall x y z:U, @Rel_of U D x y -> Strict_Rel_of U D y z -> Strict_Rel_of U D x z.
Lemma Strict_Rel_Transitive : Transitive U (Strict_Rel_of U D).