Section nested.
Variable (M : fo_model Σ' X)
(Xfin : finite_t X)
(Mdec : fo_model_dec M)
(He : forall x y, fom_rels M e (x##y##ø) <-> x = y)
(φ : nat -> X)
(HM : fol_sem M φ (Σsyms_Σnosyms ls A)).
Let HF : forall s, In s ls -> is_graph_function (fun v x => fom_rels M (inr (inl s)) (x##v)).
Let HA : fol_sem M φ (fol_rem_syms A).
End nested.

Theorem Σsyms_Σnosyms_complete :
@fo_form_fin_dec_eq_SAT_in (Σnosyms Σ) e eq_refl (Σsyms_Σnosyms ls A) X
-> fo_form_fin_discr_dec_SAT_in A X.