(* End of proof of lem_irrefl_not_refl                                     *)


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(* Symmetrie & antisymetrie                                                *)
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Definition symmetry (r a : E) (rRel : rel_sig r a a) :=
  forall x y : E, In (couple x y) r -> In (couple y x) r.Definition antisymmetry (r a : E) (rRel : rel_sig r a a) :=
  forall x y : E, In (couple x y) r -> In (couple y x) r -> x = y.

(* Specifications: *)
Lemma lem_sym_so_rdom_eq_rImg : forall (r a : E) (rRel : rel_sig r a a), symmetry r a rRel -> rdom r a a rRel = rImg r a a rRel.