unfold Symmetric, Negation in |- *.
intros U R H' x y H'0; red in |- *; intro H'1; auto.
Qed.
Hint Resolve Sym_imp_NegSym.
Definition Irreflexive (U : Set) (R : Relation U) : Prop :=
forall x : U, ~ R x x.
Definition Separating (U : Set) (R : Relation U) : Prop :=
forall x y z : U, R x y -> R x z \/ R y z.
Inductive Apartness (U : Set) (R : Relation U) : Prop :=
Definition_of_apartness :
Irreflexive U R -> Separating U R -> Apartness U R.
Hint Unfold Reflexive Irreflexive Symmetric Transitive.
Hint Resolve Definition_of_equivalence Definition_of_apartness.
(****************************************************************)
Parameter Point : Set.
Parameter Line : Set.
Parameter DiPt : Point -> Point -> Prop.
Parameter DiLn : Line -> Line -> Prop.
Parameter ConLn : Line -> Line -> Prop.

(* Specifications: *)
Axiom apart_dipt : Apartness Point DiPt.
Axiom apart_diln : Apartness Line DiLn.
Axiom apart_con : Apartness Line ConLn.
Hint Resolve apart_dipt apart_diln apart_con.