reserve X for non empty set,
  S for SigmaField of X,
  M for sigma_Measure of S,
  E for Element of S,
  F,G for Functional_Sequence of X,ExtREAL,
  I for ExtREAL_sequence,
  f,g for PartFunc of X,ExtREAL,
  seq, seq1, seq2 for ExtREAL_sequence,
  p for ExtReal,
  n,m for Nat,
  x for Element of X,
  z,D for set;

theorem
  E c= dom f & f is nonnegative & f is E-measurable & F is additive &
(for n holds F.n is_simple_func_in S & F.n is nonnegative & E c= dom(F.n)) & (
  for x st x in E holds F#x is summable & f.x = Sum(F#x)) implies ex I be
  ExtREAL_sequence st (for n holds I.n = Integral(M,(F.n)|E)) & I is summable &
  Integral(M,f|E) = Sum I
proof
  assume that
A1: E c= dom f and
A2: f is nonnegative and
A3: f is E-measurable and
A4: F is additive and
A5: for n holds F.n is_simple_func_in S & F.n is nonnegative & E c= dom
  (F.n) and
A6: for x st x in E holds F#x is summable & f.x = Sum(F#x);
  per cases;
  suppose
    E = {};
    hence thesis by A1,A3,A5,Lm2;
  end;
  suppose
A7: E <> {};
    for n be Nat holds F.n is_simple_func_in S & F.n is
    nonnegative & E c= dom(F.n) by A5;
    then E common_on_dom F by A7,SEQFUNC:def 9;
    hence thesis by A1,A2,A3,A4,A5,A6,Lm3;
  end;
end;
