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