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 Th50:
  for X,Y be non empty Subset of REAL st
  X is bounded_above & Y is bounded_above holds
  upper_bound(X++Y) = upper_bound X + upper_bound Y
proof
  let X,Y be non empty 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;
A4: (--X) is bounded_below by A1,MEASURE6:41;
  then lower_bound(--X++--Y) = lower_bound(--X)+lower_bound(--Y)
    by A3,SEQ_4:125; then
A5: lower_bound(--X++--Y) = -upper_bound(----X)+lower_bound(--Y)
      by A4,MEASURE6:43
    .= -upper_bound X + -upper_bound(----Y) by A3,MEASURE6:43
    .= -(upper_bound X + upper_bound Y);
A6: (--X)++(--Y)=--(X++Y) by Th48;
then A7: --(X++Y) is bounded_below by A4,A3,SEQ_4:124;
  reconsider XY = X++Y as Subset of REAL by MEMBERED:3;
  - upper_bound(-- --XY)= - (upper_bound X + upper_bound Y)
    by A6,A5,A7,MEASURE6:43; then
  upper_bound XY = upper_bound X + upper_bound Y;
  hence thesis;
end;
