Definition Rule_U3 (nc pc : clause) :=
 match nc , pc with
   PosSpaceClause Gamma Delta ((Lseg x y)::Sigma) ,
   NegSpaceClause Gamma' (Lseg x' Nil::Sigma') Delta' =>
     match expr_cmp x x' with
      Eq => Some (NegSpaceClause Gamma' (Lseg x y::Lseg y Nil::Sigma') Delta')
     |_ => None
     end
  | _ , _ => None
 end.

Lemma U5: forall x y z w sigma1 sigma s (K: space_denote (Lseg x y :: Lseg z w :: sigma1) s) (K': space_denote (Lseg x y:: Lseg y z :: sigma) s), pure_atom_denote (Eqv z w) s \/ space_denote (Lseg x z :: sigma) s.