let D be non empty Subset of (TOP-REAL 2); :: thesis: ( D is bounded implies E-bound D = E-bound (Cl D) )
A1: D c= Cl D by PRE_TOPC:18;
assume A2: D is bounded ; :: thesis: E-bound D = E-bound (Cl D)
then Cl D is compact by Th72;
then proj1 .: (Cl D) is bounded_above ;
then proj1 .: D is bounded_above by A1, RELAT_1:123, XXREAL_2:43;
then A3: upper_bound (proj1 .: D) = upper_bound (Cl (proj1 .: D)) by Th67
.= upper_bound (proj1 .: (Cl D)) by A2, Th76 ;
E-bound D = upper_bound (proj1 .: D) by SPRECT_1:46;
hence E-bound D = E-bound (Cl D) by A3, SPRECT_1:46; :: thesis: verum