reserve X for set,
  Y for non empty set;
reserve n for Nat;
reserve r for Real,
  M for non empty MetrSpace;

theorem
  n>= 1 implies UBD {}TOP-REAL n = [#]TOP-REAL n
proof
  set X = {B where B is Subset of TOP-REAL n: B is_outside_component_of {}
  TOP-REAL n};
  assume n>= 1;
  then
A1: [#](TOP-REAL n) is not bounded by JORDAN2C:35;
  thus UBD {}TOP-REAL n c= [#]TOP-REAL n;
  [#](TOP-REAL n) is a_component;
  then
A2: [#](the TopStruct of TOP-REAL n) is a_component by CONNSP_1:45;
  (TOP-REAL n)| [#]TOP-REAL n = the TopStruct of TOP-REAL n by TSEP_1:93;
  then
A3: [#]TOP-REAL n is_a_component_of [#]TOP-REAL n by A2,CONNSP_1:def 6;
  [#]TOP-REAL n = ({}TOP-REAL n)`;
  then [#]TOP-REAL n is_outside_component_of {}TOP-REAL n by A1,A3,
JORDAN2C:def 3;
  then [#]TOP-REAL n in X;
  then [#]TOP-REAL n c= union X by ZFMISC_1:74;
  hence thesis by JORDAN2C:def 5;
end;
