let D be non empty Subset of (TOP-REAL 2); :: thesis: ( D is Bounded implies E-bound D = E-bound (Cl D) )
assume A1: D is Bounded ; :: thesis: E-bound D = E-bound (Cl D)
then A2: Cl D is compact by Th71, Th88;
A3: ( E-bound D = sup (proj1 .: D) & E-bound (Cl D) = sup (proj1 .: (Cl D)) ) by SPRECT_1:51;
A4: proj1 .: (Cl D) is bounded_above by A2, SPRECT_1:47;
D c= Cl D by PRE_TOPC:48;
then proj1 .: D is bounded_above by A4, RELAT_1:156, XXREAL_2:43;
then sup (proj1 .: D) = sup (Cl (proj1 .: D)) by Th78
.= sup (proj1 .: (Cl D)) by A1, Th92 ;
hence E-bound D = E-bound (Cl D) by A3; :: thesis: verum