Section FSAT_ext.
Variable (Σ : fo_signature) (X : Type) (P Q : X -> Prop)
(HPQ : forall x, P x <-> Q x)
(HP : forall x (H1 H2 : P x), H1 = H2)
(HQ : forall x (H1 H2 : Q x), H1 = H2)
(M : fo_model Σ (sig P))
(Mdec : fo_model_dec M)
(Mfin : finite_t (sig P)).
Let f : sig P -> sig Q.
Let Hgf : forall x, g (f x) = x.
Let sigQ_fin : finite_t (sig Q).
Let M' : fo_model Σ (sig Q).
Variable (A : fol_form Σ).
Let ls := fol_syms A.
Let lr := fol_rels A.
Let p : @fo_projection Σ ls lr _ M _ M'.
End FSAT_ext.

Theorem FSAT_in_ext Σ A X (P Q : X -> bool) :
(forall x, P x = true <-> Q x = true)
-> @fo_form_fin_dec_SAT_in Σ A (sig (fun x => P x = true))
<-> @fo_form_fin_dec_SAT_in Σ A (sig (fun x => Q x = true)).