let D be non empty Subset of (TOP-REAL 2); :: thesis: ( D is Bounded implies W-bound D = W-bound (Cl D) )
assume A1: D is Bounded ; :: thesis: W-bound D = W-bound (Cl D)
then A2: Cl D is compact by Th71, Th88;
A3: ( W-bound D = inf (proj1 .: D) & W-bound (Cl D) = inf (proj1 .: (Cl D)) ) by SPRECT_1:48;
A4: proj1 .: (Cl D) is bounded_below by A2, SPRECT_1:46;
D c= Cl D by PRE_TOPC:48;
then proj1 .: D is bounded_below by A4, RELAT_1:156, XXREAL_2:44;
then inf (proj1 .: D) = inf (Cl (proj1 .: D)) by Th77
.= inf (proj1 .: (Cl D)) by A1, Th92 ;
hence W-bound D = W-bound (Cl D) by A3; :: thesis: verum