prolog
[ sym_inter isGram2 inter_union isGram4 inmon_Der_imp_Der_d2
inmon_Der_imp_inmon_R2_d ] 15.
(*Intros N_dans_X N_dans_V1 N_dans_V2 u v G_R1 G_R2 inter_V1_V2_empty inmon_X_V2_u Der_Ru_u_v. Apply inmon_Der_imp_inmon_R2_d with u. Apply isGram4 with S2;Assumption.  Apply inmon_Der_imp_Der_d2.  Assumption.  Assumption.  Assumption.  Apply isGram4 with S1;Assumption.  Apply isGram4 with S2;Assumption.  Apply inter_union. Apply isGram2 with R1 S1; Assumption. Apply sym_inter;Assumption.  Assumption.  Assumption.  Assumption.*)
Qed.

(* Specifications: *)
Lemma Gunion_disj_Derivestar2 :
~ dans S X ->
~ dans S V1 ->
~ dans S V2 ->
isGram X V1 R1 S1 ->
isGram X V2 R2 S2 ->
inter V1 V2 empty ->
forall u v : Word,
Derivestar Ru u v -> inmonoid (union X V2) u -> Derivestar R2 u v.