theorem
  D is bounded implies S-bound D = S-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_below;
  then proj2.:D is bounded_below by A1,RELAT_1:123,XXREAL_2:44;
  then
A3: lower_bound (proj2.:D) = lower_bound Cl(proj2.:D) by Th66
    .= lower_bound (proj2.:Cl D) by A2,Th77;
  S-bound D = lower_bound (proj2.:D) by SPRECT_1:44;
  hence thesis by A3,SPRECT_1:44;
end;
