lemma lub_compat: "lub A x = isLub UNIV A x"

 lemma real_complete: "\<exists>a::real. a \<in> A \<Longrightarrow> \<exists>y. \<forall>a \<in> A. a \<le> y \<Longrightarrow> \<exists>x. lub A x"

 proof -

   by (intro exI[of _ "Sup A"]) (auto intro!: cSup_upper cSup_least simp: lub_def)

   have "isUb UNIV A = (\<lambda>x. A *<= x \<and> x \<in> UNIV)"

     by (rule ext) (simp only: isUb_def)

   then show ?thesis

     by (simp only: lub_def isLub_def leastP_def setge_def setle_def) blast

 qed

 lemma real_complete:

   fixes A :: "real set"

   assumes nonempty: "\<exists>a. a \<in> A"

     and ex_upper: "\<exists>y. \<forall>a \<in> A. a \<le> y"

   shows "\<exists>x. lub A x"

 proof -

   from ex_upper have "\<exists>y. isUb UNIV A y"

     unfolding isUb_def setle_def by blast

   with nonempty have "\<exists>x. isLub UNIV A x"

     by (rule reals_complete)

   then show ?thesis by (simp only: lub_compat)

 qed