Section nested.
Variables (A : fol_form (Σrel n))
(X : Type)
(M2 : fo_model (Σrel 2) X)
(M2fin : finite_t X)
(M2dec : fo_model_dec M2)
(ψ : nat -> X)
(HA : fol_sem M2 ψ (Σn_Σ2_enc A)).
Let mem a b := fom_rels M2 tt (a##b##ø).
Let mem_dec : forall x y, { mem x y } + { ~ mem x y }.
Let HP1 : finite_t (sig P).
Let Mn : fo_model (Σrel n) (sig P).
End nested.

Theorem SAT2_SATn A : fo_form_fin_dec_SAT (@Σn_Σ2_enc n A)
-> fo_form_fin_dec_SAT A.