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.

(*The axiom schema for U4 from Figure 2, used as an auxiliary lemma for the proof of U4*)
Lemma U4_axiomSchema: forall x y z w Sigma s,
space_denote (Lseg x y::Lseg y z::Next z w::Sigma) s ->
space_denote (Lseg x z::Next z w::Sigma) s.