let D be non empty Subset of (TOP-REAL 2); :: thesis: ( D is Bounded implies N-bound D = N-bound (Cl D) )
assume A1: D is Bounded ; :: thesis: N-bound D = N-bound (Cl D)
then A2: Cl D is compact by Th71, Th88;
A3: ( N-bound D = sup (proj2 .: D) & N-bound (Cl D) = sup (proj2 .: (Cl D)) ) by SPRECT_1:50;
A4: proj2 .: (Cl D) is bounded_above by A2, SPRECT_1:47;
D c= Cl D by PRE_TOPC:48;
then proj2 .: D is bounded_above by A4, RELAT_1:156, XXREAL_2:43;
then sup (proj2 .: D) = sup (Cl (proj2 .: D)) by Th78
.= sup (proj2 .: (Cl D)) by A1, Th93 ;
hence N-bound D = N-bound (Cl D) by A3; :: thesis: verum