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.Inductive Apartness (U : Set) (R : Relation U) : Prop :=
    Definition_of_apartness :
      Irreflexive U R -> Separating U R -> Apartness U R.(****************************************************************)

Parameter Point : Set.Parameter Line : Set.Parameter DiPt : Point -> Point -> Prop.Parameter DiLn : Line -> Line -> Prop.Parameter ConLn : Line -> Line -> Prop.Axiom apart_dipt : Apartness Point DiPt.Axiom apart_diln : Apartness Line DiLn.Axiom apart_con : Apartness Line ConLn.

(* Specifications: *)
Theorem Apart_imp_Sym : forall (U : Set) (R : Relation U), Apartness U R -> Symmetric U R.
Theorem sym_DiPt : forall x y : Point, DiPt x y -> DiPt y x.
Theorem sym_DiLn : forall x y : Line, DiLn x y -> DiLn y x.
Theorem sym_ConLn : forall x y : Line, ConLn x y -> ConLn y x.
Theorem Neg_apart_equiv : forall (U : Set) (R : Relation U), Apartness U R -> Equivalence U (Negation U R).