theorem
  f|A is bounded implies rng upper_sum_set(f) is bounded_below
proof
  set D1 = the Division of A;
  assume
A1: f|A is bounded;
 take lower_sum(f,D1);
    let a be ExtReal;
    assume a in rng upper_sum_set(f);
    then consider D be Element of divs A such that
A2: D in dom upper_sum_set(f) & a = (upper_sum_set(f)).D by PARTFUN1:3;
    reconsider D as Division of A by INTEGRA1:def 3;
    a = upper_sum(f,D) by A2,INTEGRA1:def 10;
    hence thesis by A1,INTEGRA1:48;
end;
