theorem
  2 <= n implies for A, B, P being Subset of TOP-REAL n st P is bounded
  & A is_outside_component_of P & B is_outside_component_of P holds A = B
proof
  assume
A1: 2 <= n;
  let A, B, P be Subset of TOP-REAL n such that
A2: P is bounded and
A3: A is_outside_component_of P and
A4: B is_outside_component_of P;
A5: B is_a_component_of P` by A4;
  UBD P is_outside_component_of P by A1,A2,Th53;
  then
A6: UBD P is_a_component_of P`;
A7: P` is non empty by A1,A2,Th51,XXREAL_0:2;
A8: B <> {} by A4;
A9: B c= UBD P by A4,Th14;
A10: A c= UBD P by A3,Th14;
A11: A is_a_component_of P` by A3;
  then A <> {} by A7,SPRECT_1:4;
  then A = UBD P by A11,A6,A10,GOBOARD9:1,XBOOLE_1:69;
  hence thesis by A5,A8,A6,A9,GOBOARD9:1,XBOOLE_1:69;
end;
