(* ------------------------------------------------------------- *)

Lemma Par_open_tm_wrt_tm_preservation: forall G D B1 B2 x, Par G D (open_tm_wrt_tm B1 (a_Var_f x)) B2 -> exists B', B2 = open_tm_wrt_tm B' (a_Var_f x) /\ Par G D (open_tm_wrt_tm B1 (a_Var_f x)) (open_tm_wrt_tm B' (a_Var_f x)).