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;
