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:18;
assume A2: D is bounded ; :: thesis: N-bound D = N-bound (Cl D)
then Cl D is compact by Th72;
then proj2 .: (Cl D) is bounded_above ;
then proj2 .: D is bounded_above by A1, RELAT_1:123, XXREAL_2:43;
then A3: upper_bound (proj2 .: D) = upper_bound (Cl (proj2 .: D)) by Th67
.= upper_bound (proj2 .: (Cl D)) by A2, Th77 ;
N-bound D = upper_bound (proj2 .: D) by SPRECT_1:45;
hence N-bound D = N-bound (Cl D) by A3, SPRECT_1:45; :: thesis: verum