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;
