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_below & 0<=r holds
    r**X is bounded_below
proof
  let X be non empty Subset of REAL;
  assume that
A1: X is bounded_below and
A2: 0<=r;
  consider b be Real such that
A3: b is LowerBound of X by A1;
A4: for x be Real st x in X holds b <= x by A3,XXREAL_2:def 2;
  r*b is LowerBound of r**X
  proof
    let y be ExtReal;
    assume y in r**X;
    then y in {r*x : x in X} by Th8;
    then ex x st y=r*x & x in X;
    hence thesis by A2,A4,XREAL_1:64;
  end;
  hence thesis;
end;
