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

theorem
  for P being connected Subset of TOP-REAL n, Q being Subset of TOP-REAL
  n st P misses Q holds P c= UBD Q or P c= BDD Q
proof
  let P be connected Subset of TOP-REAL n, Q being Subset of TOP-REAL n such
  that
A1: P misses Q;
  per cases;
  suppose
    P is empty;
    hence thesis;
  end;
  suppose
    Q = [#]TOP-REAL n;
    then P = {} by A1,XBOOLE_1:67;
    hence thesis;
  end;
  suppose that
A2: P is not empty and
 Q <> [#]TOP-REAL n;
    P c= Q` by A1,SUBSET_1:23;
    then consider C being Subset of TOP-REAL n such that
A3: C is_a_component_of Q` and
A4: P c= C by A2,GOBOARD9:3;
    C is_inside_component_of Q or C is_outside_component_of Q by A3,Th14;
    then C c= UBD Q or C c= BDD Q by JORDAN2C:22,23;
    hence thesis by A4;
  end;
end;
