unfold pure_bound. destruct a.
split.
simpl.
apply expr_bound_more with (var_upto (freshmax_expr e)).
apply included_var_upto.  apply Ile_var_max1.
apply expr_bound_freshmax.
simpl.
apply expr_bound_more with (var_upto (freshmax_expr e0)).
apply included_var_upto.  apply Ile_var_max2.
apply expr_bound_freshmax.
Qed.

Lemma space_bound_freshmax:
forall a, space_bound (var_upto (freshmax_space_atom a)) a.