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 BDD {}TOP-REAL n = {}TOP-REAL n
proof
  set X = {B where B is Subset of TOP-REAL n: B is_inside_component_of {}(
  TOP-REAL n)};
  assume n>= 1;
  then
A1: [#](TOP-REAL n) is not bounded by JORDAN2C:35;
  now
    [#](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;
    assume X <> {};
    then consider x being object such that
A4: x in X by XBOOLE_0:def 1;
    consider B being Subset of TOP-REAL n such that
    x = B and
A5: B is_inside_component_of {}(TOP-REAL n) by A4;
    B is_a_component_of ({}(TOP-REAL n))` by A5,JORDAN2C:def 2;
    then B = [#]TOP-REAL n by A3,Th7;
    hence contradiction by A1,A5,JORDAN2C:def 2;
  end;
  hence thesis by JORDAN2C:def 4,ZFMISC_1:2;
end;
