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

theorem Th14:
  for X be non empty Subset of REAL st X is bounded_above & r<=0
  holds lower_bound(r**X) = r*(upper_bound X)
proof
  let X be non empty Subset of REAL;
  assume that
A1: X is bounded_above and
A2: r<=0;
A3: for a be Real st a in r**X holds r*(upper_bound X) <= a
  proof
    let a be Real;
    assume a in r**X;
    then a in {r*x : x in X} by Th8;
    then consider x such that
A4: a=r*x and
A5: x in X;
    x <= upper_bound X by A1,A5,SEQ_4:def 1;
    hence thesis by A2,A4,XREAL_1:65;
  end;
 for b be Real st for a be Real st a in r**X holds a >= b
  holds r*(upper_bound X) >= b
  proof
    consider x being Element of REAL such that
A6: x in X by SUBSET_1:4;
    let b be Real;
    assume
A7: for a be Real st a in r**X holds a >= b;
    reconsider x as Real;
    r*x in {r*y : y in X} by A6;
    then
A8: r*x in r**X by Th8;
    now
      per cases by A2;
      suppose
        r=0;
        hence thesis by A7,A8;
      end;
      suppose
A9:    r<0;
        for z be Real st z in X holds z <= b/r
        proof
          let z be Real;
          assume z in X;
          then r*z in {r*y : y in X};
          then r*z in r**X by Th8;
          hence thesis by A7,A9,XREAL_1:80;
        end;
        then upper_bound X <= b/r by SEQ_4:45;
        then r*(upper_bound X) >= b/r*r by A9,XREAL_1:65;
        hence thesis by A9,XCMPLX_1:87;
      end;
    end;
    hence thesis;
  end;
  hence thesis by A3,SEQ_4:44;
end;
