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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.
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