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;
reserve f for bounded integrable Function of I,REAL;
reserve jauge for positive-yielding Function of I,REAL;
reserve D for tagged_division of I;
reserve r1,r2,s for Real,
           D,D1 for Division of I,
             fc for Function of I,REAL;

theorem Th43:
  ex i being Nat st i in dom D &
  min rng upper_volume(fc,D) = (upper_volume(fc,D)).i
  proof
    inf rng upper_volume(fc,D) in rng upper_volume(fc,D) by XXREAL_2:def 5;
    then consider x be object such that
A1: x in dom upper_volume(fc,D) and
A2: (upper_volume(fc,D)).x = inf rng upper_volume(fc,D) by FUNCT_1:def 3;
A3: dom upper_volume(fc,D)
      = Seg len upper_volume(fc,D) by FINSEQ_1:def 3
     .= Seg len D by INTEGRA1:def 6
     .= dom D by FINSEQ_1:def 3;
    reconsider i = x as Nat by A1;
    take i;
    thus thesis by A2,A3,A1;
  end;
