Definition Rule_U1 (nc pc : clause) :=
 match nc , pc with
   PosSpaceClause Gamma Delta ((Next x z)::Sigma) ,
   NegSpaceClause Gamma' (Lseg x' z'::Sigma') Delta' =>
     match expr_cmp x x', expr_cmp z z' with
      Eq, Eq => Some (NegSpaceClause Gamma' (Next x' z'::Sigma')
                        (Eqv x z :: Delta'))
     | _, _ => None
     end
  | _ , _ => None
 end.

Lemma U1_axiomSchema: forall x z Sigma s, space_denote (Next x z::Sigma) s -> pure_atom_denote (Eqv x z) s \/ space_denote (Lseg x z::Sigma) s.