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diff --git a/theories/Relations/Relations.v b/theories/Relations/Relations.v
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+(************************************************************************)
+(*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
+(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *)
+(*   \VV/  **************************************************************)
+(*    //   *      This file is distributed under the terms of the       *)
+(*         *       GNU Lesser General Public License Version 2.1        *)
+(************************************************************************)
+
+(*i $Id: Relations.v,v 1.6.2.1 2004/07/16 19:31:16 herbelin Exp $ i*)
+
+Require Export Relation_Definitions.
+Require Export Relation_Operators.
+Require Export Operators_Properties.
+
+Lemma inverse_image_of_equivalence :
+ forall (A B:Set) (f:A -> B) (r:relation B),
+   equivalence B r -> equivalence A (fun x y:A => r (f x) (f y)).
+intros; split; elim H; red in |- *; auto.
+intros _ equiv_trans _ x y z H0 H1; apply equiv_trans with (f y); assumption.
+Qed.
+
+Lemma inverse_image_of_eq :
+ forall (A B:Set) (f:A -> B), equivalence A (fun x y:A => f x = f y).
+split; red in |- *;
+ [  (* reflexivity *) reflexivity
+ |  (* transitivity *) intros; transitivity (f y); assumption
+ |  (* symmetry *) intros; symmetry  in |- *; assumption ].
+Qed.
+\ No newline at end of file