theories/Relations

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+
+(* $Id$ *)
+
+Require Export Relation_Definitions.
+Require Export Relation_Operators.
+Require Export Operators_Properties.
+
+Lemma inverse_image_of_equivalence : (A,B:Set)(f:A->B)
+  (r:(relation B))(equivalence B r)->(equivalence A [x,y:A](r (f x) (f y))).
+Intros; Split; Elim H; Red; Auto.
+Intros; Apply equiv_trans with (f y); Assumption.
+Save.
+
+Lemma inverse_image_of_eq : (A,B:Set)(f:A->B)
+  (equivalence A [x,y:A](f x)=(f y)).
+Split; Red;
+[ (* reflexivity *)  Reflexivity
+| (* transitivity *) Intros; Transitivity (f y); Assumption
+| (* symmetry *) Intros; Symmetry; Assumption
+].
+Save.