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:18;
assume A2: D is bounded ; :: thesis: W-bound D = W-bound (Cl D)
then Cl D is compact by Th72;
then proj1 .: (Cl D) is bounded_below ;
then proj1 .: D is bounded_below by A1, RELAT_1:123, XXREAL_2:44;
then A3: lower_bound (proj1 .: D) = lower_bound (Cl (proj1 .: D)) by Th66
.= lower_bound (proj1 .: (Cl D)) by A2, Th76 ;
W-bound D = lower_bound (proj1 .: D) by SPRECT_1:43;
hence W-bound D = W-bound (Cl D) by A3, SPRECT_1:43; :: thesis: verum