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 Th34:
  F is additive & F is with_the_same_dom & dom f c= dom(F.0) & x
in dom f & F#x is summable & f.x = Sum(F#x) implies f.x = lim((Partial_Sums F)#
  x)
proof
  set PF = Partial_Sums F;
  assume that
A1: F is additive and
A2: F is with_the_same_dom and
A3: dom f c= dom(F.0) and
A4: x in dom f and
A5: F#x is summable and
A6: f.x = Sum(F#x);
  set PFx = Partial_Sums(F#x);
  PFx is convergent by A5;
  then
A7: PF#x is convergent by A1,A2,A3,A4,Th33;
  per cases by A7,MESFUNC5:def 12;
  suppose
A8: ex g be Real st lim(PF#x) = g & (for p be Real st 0<p
ex n be Nat st for m be Nat st n<=m holds |. (PF#x).m- lim (PF#x) .| < p) & (PF
    #x) is convergent_to_finite_number;
    then
A9: PFx is convergent_to_finite_number by A1,A2,A3,A4,Th33;
    then
A10: not PFx is convergent_to_+infty by MESFUNC5:50;
A11: not PFx is convergent_to_-infty by A9,MESFUNC5:51;
    PFx is convergent by A9;
    then
A12: ex g be Real st f.x = g & (for p be Real st 0<p ex n be
    Nat st for m be Nat st n<=m holds |. PFx.m- f.x .| < p) & PFx is
    convergent_to_finite_number by A6,A10,A11,MESFUNC5:def 12;
    now
      let p be Real;
      assume 0<p;
      then consider n be Nat such that
A13:  for m be Nat st n<=m holds |. PFx.m - f.x .| < p by A12;
      take n;
      let m be Nat;
      assume
A14:  n<=m;
      PFx.m = (PF#x).m by A1,A2,A3,A4,Th32;
      hence |. (PF#x).m - f.x .| < p by A13,A14;
    end;
    hence thesis by A7,A8,A12,MESFUNC5:def 12;
  end;
  suppose
A15: lim(PF#x)=+infty & PF#x is convergent_to_+infty;
    then
A16: PFx is convergent_to_+infty by A1,A2,A3,A4,Th33;
    then PFx is convergent;
    hence thesis by A6,A15,A16,MESFUNC5:def 12;
  end;
  suppose
A17: lim(PF#x)=-infty & PF#x is convergent_to_-infty;
    then
A18: PFx is convergent_to_-infty by A1,A2,A3,A4,Th33;
    then PFx is convergent;
    hence thesis by A6,A17,A18,MESFUNC5:def 12;
  end;
end;
