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 upper_integrable holds
  ex r being Real st for D being Division of I holds r < upper_sum(f,D)
  proof
    let f be PartFunc of I,REAL;
    assume f is upper_integrable;
    then rng upper_sum_set(f) is bounded_below by INTEGRA1:def 12;
    then consider r be Real such that
A1: for y being object st y in dom upper_sum_set(f) holds
      r < (upper_sum_set(f)).y by INTEGRA1:12,SEQ_2:def 2;
A2: dom upper_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 r < (upper_sum_set(f)).D by A1,A2;
    hence thesis by INTEGRA1:def 10;
  end;
