Section build_the_model.
Let l := proj1_sig fin.
Let Hl : forall x, In x l := proj2_sig fin.
Let Q : fin_quotient fom_eq.
Let H2 s v : In s lr -> fom_rels M s v <-> fom_rels Md s (vec_map cls v).
Let f : fo_projection ls lr M Md.
(* Every finite & decidable model can be projected to pos n with decidable relations and such that identity is exactly FO undistinguishability *)

Theorem fo_fin_model_discretize :
{ n : nat &
{ Md : fo_model Σ (pos n) &
{ _ : fo_model_dec Md &
{ _ : fo_projection ls lr M Md &
(forall p q, fo_bisimilar Md p q <-> p = q) } } } }.