Definition Rule_U2 (nc pc : clause) :=
 match nc , pc with
   PosSpaceClause Gamma Delta ((Next x y)::Sigma) ,
   NegSpaceClause Gamma' (Lseg x' z::Sigma') Delta' =>
     match expr_cmp x x' with
      Eq => Some (NegSpaceClause Gamma' (Next x y::Lseg y z::Sigma')
                   (Eqv x z :: Delta'))
     |_ => None
     end
  | _ , _ => None
 end.

Lemma U2_axiomSchema: forall x y z Sigma s, space_denote (Next x y::Lseg y z::Sigma) s -> pure_atom_denote (Eqv x z) s \/ space_denote (Lseg x z::Sigma) s.