reserve Z for set;

theorem Th9:
  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 st f is
  bounded & f is integrable & delta(T) is convergent & lim delta(T)=0 holds
  middle_sum(f,S) is convergent & lim (middle_sum(f,S))=integral(f)
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;
  assume that
A1: f is bounded and
A2: f is integrable and
A3: delta(T) is convergent & lim delta(T)=0;
  f|A is bounded_below by A1;
  then
A5: for i be Element of NAT holds (lower_sum(f,T)).i <= (middle_sum(f,S)).i
  by Th5;
A6: f|A= f;
  then
A7: lower_sum(f,T) is convergent & upper_sum(f,T) is convergent by A1,A3,
INTEGRA4:8,9;
  f|A is bounded_above by A1;
  then
A8: for i be Element of NAT holds (middle_sum(f,S)).i <= (upper_sum(f,T)).i
  by Th6;
A9: lim upper_sum(f,T)-lim lower_sum(f,T)=0 by A1,A2,A3,A6,INTEGRA4:12;
  then lim lower_sum(f,T) = integral(f) by A1,A3,A6,INTEGRA4:9;
  hence thesis by A7,A9,A5,A8,Lm5;
end;
