reserve a,a1,b,b1,x,y for Real,
  F,G,H for FinSequence of REAL,
  i,j,k,n,m for Element of NAT,
  I for Subset of REAL,
  X for non empty set,
  x1,R,s for set;
reserve A for non empty closed_interval Subset of REAL;
reserve A, B for non empty closed_interval Subset of REAL;
reserve r for Real;
reserve D, D1, D2 for Division of A;
reserve f, g for Function of A,REAL;

theorem Th49:
  for X,Y being Subset of REAL st X is bounded_above & Y is
  bounded_above holds X++Y is bounded_above
proof
  let X,Y be Subset of REAL;
  assume that
A1: X is bounded_above and
A2: Y is bounded_above;
A3: (--Y) is bounded_below by A2,MEASURE6:41;
  (--X) is bounded_below by A1,MEASURE6:41;
  then
A4: (--X)++(--Y) is bounded_below by A3,SEQ_4:124;
  reconsider XY = X++Y as Subset of REAL by MEMBERED:3;
  --XY is bounded_below by Th48,A4;
  hence thesis by MEASURE6:41;
end;
