reserve Z for set;

theorem Th6:
  for A be non empty closed_interval Subset of REAL, f being Function of A,
  REAL, T being DivSequence of A, S be middle_volume_Sequence of f,T, i be
Element of NAT st f|A is bounded_above holds (middle_sum(f,S)).i <= (upper_sum(
  f,T)).i
proof
  let A be non empty closed_interval Subset of REAL,
  f being Function of A,REAL, T being
  DivSequence of A, S be middle_volume_Sequence of f,T, i be Element of NAT;
  assume
A1: f|A is bounded_above;
  (middle_sum(f,S)).i= middle_sum(f,S.i) & (upper_sum(f,T)).i = upper_sum(
  f,T. i) by Def4,INTEGRA2:def 2;
  hence thesis by A1,Th2;
end;
