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
| Intros; Transitivity (f y); Assumption
| Intros; Symmetry; Assumption
].
Save
.