let D be non empty Subset of (TOP-REAL 2); :: thesis: ( D is Bounded implies N-bound D = N-bound (Cl D) )
A1: D c= Cl D by PRE_TOPC:48;
assume A2: D is Bounded ; :: thesis: N-bound D = N-bound (Cl D)
then Cl D is compact by Th71, Th88;
then proj2 .: (Cl D) is bounded_above by SPRECT_1:47;
then proj2 .: D is bounded_above by A1, RELAT_1:156, XXREAL_2:43;
then A3: sup (proj2 .: D) = sup (Cl (proj2 .: D)) by Th78
.= sup (proj2 .: (Cl D)) by A2, Th93 ;
N-bound D = sup (proj2 .: D) by SPRECT_1:50;
hence N-bound D = N-bound (Cl D) by A3, SPRECT_1:50; :: thesis: verum