reserve a,b,c,d,e for Real;
reserve X,Y for set,
          Z for non empty set,
          r for Real,
          s for ExtReal,
          A for Subset of REAL,
          f for real-valued Function;
reserve I for non empty closed_interval Subset of REAL,
       TD for tagged_division of I,
        D for Division of I,
        T for Element of set_of_tagged_Division(D),
        f for PartFunc of I,REAL;
reserve f for Function of I,REAL;
reserve f,g for HK-integrable Function of I,REAL,
          r for Real;

theorem
  for f being PartFunc of I,REAL st f is lower_integrable holds
  ex r being Real st for D being Division of I holds lower_sum(f,D) < r
  proof
    let f be PartFunc of I,REAL;
    assume f is lower_integrable;
    then rng lower_sum_set(f) is bounded_above by INTEGRA1:def 13;
    then consider r be Real such that
A1: for y being object st y in dom lower_sum_set(f) holds
      (lower_sum_set(f)).y < r by INTEGRA1:14,SEQ_2:def 1;
A2: dom lower_sum_set(f) = divs I by FUNCT_2:def 1;
    take r;
    let D be Division of I;
    D in divs(I) by INTEGRA1:def 3;
    then (lower_sum_set(f)).D < r by A1,A2;
    hence thesis by INTEGRA1:def 11;
  end;
