unfold space_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.
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 spaces_bound'_freshmax:
forall gamma,
spaces_bound' (var_upto (freshmax_list freshmax_space_atom gamma)) gamma.