by (meson A_\<alpha>\<beta> Pi_I' \<alpha>_sub inv_into_into subset_eq)
then have fg: "f \<circ> (\<lambda>X. ?g ` X) \<in> [elts \<alpha>]\<^bsup>2\<^esup> \<rightarrow> {..<2}"
by (rule nsets_compose_image_funcset [OF f _ inj_g])
have g_less: "?g x < ?g y" if "x < y" "x \<in> elts \<alpha>" "y \<in> elts \<alpha>" for x y
using Pi_mem [OF g]
by (meson A_\<alpha>\<beta> Ord_in_Ord Ord_not_le ord \<open>small A\<close> dual_order.trans elts_subset_ON inv_ordermap_VWF_mono_le ot that vsubsetD)
obtain i H where "i < 2" "H \<subseteq> elts \<alpha>"
and ot_eq: "tp H = [k,\<gamma>]!i" "(f \<circ> (\<lambda>X. ?g ` X)) ` (nsets H 2) \<subseteq> {i}"
using ii partn_lst_E [OF part fg] by (auto simp: eval_nat_numeral)
...
...
@@ -411,12 +403,7 @@ next
have gH: "?g ` H \<subseteq> elts (\<alpha> * \<beta>)"