(* with Prolog...*)
intros;
prolog
[ inmon_Der_imp_inmon_R2 sym_inter isGram2 inter_union isGram4
inmon_Der_imp_Der2 ] 5.
(* without Prolog...*)
(*Intros u v G_R1 G_R2 inter_V1_V2_empty inmon_X_V2_u Der_Ru_u_v. Apply inmon_Der_imp_inmon_R2 with u. Apply isGram4 with S2;Assumption.   Apply inmon_Der_imp_Der2. 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.

Lemma Gunion_Derivestar2 :
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.