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;

theorem Th18:
  f is lower_integrable implies lower_sum(f,D) <= lower_integral(f)
  proof
    assume f is lower_integrable;
    then
A1: rng lower_sum_set(f) is bounded_above by INTEGRA1:def 13;
    set r = lower_integral(f);
    r = upper_bound rng lower_sum_set(f) by INTEGRA1:def 15;
    then
A2: for s be Real st s in rng lower_sum_set(f) holds s <= r by A1,SEQ_4:def 1;
A3: dom lower_sum_set(f) = divs I by PARTFUN1:def 2;
    D in divs I by INTEGRA1:def 3;
    then (lower_sum_set(f)).D <= r by A3,A2,FUNCT_1:3;
    hence thesis by INTEGRA1:def 11;
  end;
