(* Contribution to the Coq Library   V6.3 (July 1999)                    *)
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(* This contribution was updated for Coq V5.10 by the COQ workgroup.        *)
(* January 1995                                                             *)
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(*                           Relations_1_facts.v                            *)
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Definition Complement (U : Type) (R : Relation U) :
Relation U := fun x y : U => ~ R x y.
unfold Symmetric, Complement in |- *.
intros U R H' x y H'0; red in |- *; intro H'1; apply H'0; auto.
Qed.

Theorem Equiv_from_preorder :
forall (U : Type) (R : Relation U),
Preorder U R -> Equivalence U (fun x y : U => R x y /\ R y x).