(* similar remark apply here… *)

Lemma BC_to_TRS_infos_subst_comp_third h rl tl st n s f:
let st' := fst (lm_to_ml (map BC_to_TRS rl) (fst (BC_to_TRS h st))) in
f ∈ flat_map all_lhs_funcs (snd (lm_to_ml (map BC_to_TRS tl) st')) →
get_info (snd ((BC_to_TRS h >>+ lm_to_ml (map BC_to_TRS rl) >>++
lm_to_ml (map BC_to_TRS tl) >>+ Mnew_fun >>== Mmc n s) st)) f =
get_info_l (snd (lm_to_ml (map BC_to_TRS tl) st')) f.