reserve a, b for Real,
  r for Real,
  rr for Real,
  i, j, n for Nat,
  M for non empty MetrSpace,
  p, q, s for Point of TOP-REAL 2,
  e for Point of Euclid 2,
  w for Point of Euclid n,
  z for Point of M,
  A, B for Subset of TOP-REAL n,
  P for Subset of TOP-REAL 2,
  D for non empty Subset of TOP-REAL 2;
reserve a, b for Real;
reserve a, b for Real;
reserve r for Real;

theorem
  for X, Y being Subset of TOP-REAL n st n >= 1 & the carrier of
  TOP-REAL n = X \/ Y & X is bounded holds Y is non bounded
proof
  set M = TOP-REAL n;
  let X, Y be Subset of M such that
A1: n >= 1 and
A2: the carrier of M = X \/ Y;
  reconsider E = [#]M as Subset of Euclid n by TOPREAL3:8;
  [#](TOP-REAL n) is non bounded by A1,JORDAN2C:35;
  then
A3: E is non bounded by JORDAN2C:11;
  reconsider D = Y as Subset of Euclid n by TOPREAL3:8;
  assume X is bounded;
  then reconsider C = X as bounded Subset of Euclid n by JORDAN2C:11;
A4: the carrier of Euclid n = C \/ D by A2,TOPREAL3:8;
  E = [#]Euclid n by TOPREAL3:8;
  then Euclid n is non bounded by A3;
  then D is non bounded by A4,Th63;
  hence thesis by JORDAN2C:11;
end;
