reserve a,b,r,x,y for Real,
  i,j,k,n for Nat,
  x1 for set;

theorem
  for X be non empty Subset of REAL st X is bounded_above & 0<=r holds
    r**X is bounded_above
proof
  let X be non empty Subset of REAL;
  assume that
A1: X is bounded_above and
A2: 0<=r;
  consider b be Real such that
A3: b is UpperBound of X by A1;
A4: for x be Real st x in X holds x <= b by A3,XXREAL_2:def 1;
  r*b is UpperBound of r**X
  proof
    let y be ExtReal;
    assume y in r**X;
    then y in {r*x : x in X} by Th8;
    then consider x such that
A5: y=r*x and
A6: x in X;
    x <= b by A4,A6;
    hence thesis by A2,A5,XREAL_1:64;
  end;
  hence thesis;
end;
