Section resultats.
Variable X V1 R1 : Ensf.
Variable S1 : Elt.
Variable V2 R2 : Ensf.
Variable S2 : Elt.
Let C := Gunion V1 R1 V2 R2.
Let Vu := fst C.
Let Ru := snd C.
intro G1.
intro G2.
unfold Vu in |- *.
unfold C in |- *.
unfold Gunion in |- *.
simpl in |- *.
apply union_inter.
apply isGram2 with R1 S1; assumption.
apply isGram2 with R2 S2; assumption.
Qed.
intros.
unfold Ru in |- *; unfold C in |- *; unfold Gunion in |- *; simpl in |- *.
apply Regles_union; unfold Vu in |- *; unfold C in |- *;
unfold Gunion in |- *; simpl in |- *.
apply Regles_V with V1;
[ auto | apply isGram4 with S1; auto ].
apply Regles_V with V2;
[ auto | apply isGram4 with S2; auto ].
Qed.

Lemma inmon_Der_imp_Der :
Regles X V1 R1 ->
Regles X V2 R2 ->
inter (union X V1) V2 empty ->
forall u v : Word, Derive Ru u v -> inmonoid (union X V1) u -> Derive R1 u v.