theorem
  D is bounded implies N-bound D = N-bound Cl D
proof
A1: D c= Cl D by PRE_TOPC:18;
  assume
A2: D is bounded;
  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 thesis by A3,SPRECT_1:45;
end;
